Springer Lecture Notes in Physics 716

7 

Field-Theory Approaches 

to Nonequilibrium Dynamics 

U. C. Täuber 

Department of Physics, Center for Stochastic Processes in Science and Engineering 

Virginia Polytechnic Institute and State University 

Blacksburg, Virginia 24061-0435, USA 

tauber@vt.edu 

It is explained how field-theoretic methods and the dynamic renormalisation 

group (RG) can be applied to study the universal scaling properties of systems 

that either undergo a continuous phase transition or display generic scale invariance, 

both near and far from thermal equilibrium. Part 1 introduces the 

response functional field theory representation of (nonlinear) Langevin equations. 

The RG is employed to compute the scaling exponents for several universality 

classes governing the critical dynamics near second-order phase transitions 

in equilibrium. The effects of reversible mode-coupling terms, quenching 

from random initial conditions to the critical point, and violating the 

detailed balance constraints are briefly discussed. It is shown how the same 

formalism can be applied to nonequilibrium systems such as driven diffusive 

lattice gases. Part 2 describes how the master equation for stochastic particle 

reaction processes can be mapped onto a field theory action. The RG is 

then used to analyse simple diffusion-limited annihilation reactions as well as 

generic continuous transitions from active to inactive, absorbing states, which 

are characterised by the power laws of (critical) directed percolation. Certain 

other important universality classes are mentioned, and some open issues are 

listed. 

7.1 Critical Dynamics 

Field-theoretic tools and the renormalisation group (RG) method have had 

a tremendous impact in our understanding of the universal power laws that 

emerge near equilibrium critical points (see, e.g., Refs. [1–6]), including the 

associated dynamic critical phenomena [7,8]. Our goal here is to similarly describe 

the scaling properties of systems driven far from thermal equilibrium, 

which either undergo a continuous nonequilibrium phase transition or display 

generic scale invariance. We are then confronted with capturing the (stochastic) 

dynamics of the long-wavelength modes of the “slow” degrees of freedom, 

U.C. Täuber: Field-Theory Approaches to Nonequilibrium Dynamics, Lect. Notes Phys. 716, 

295–348 (2007) 

DOI 10.1007/3-540-69684-9 7 c○ Springer-Verlag Berlin Heidelberg 2007

296 U. C. Täuber 

namely the order parameter for the transition, any conserved quantities, and 

perhaps additional relevant variables. In these lecture notes, I aim to briefly 

describe how a representation in terms of a field theory action can be obtained 

for (1) general nonlinear Langevin stochastic differential equations [8, 9]; and 

(2) for master equations governing classical particle reaction–diffusion systems 

[10–12]. I will then demonstrate how the dynamic (perturbative) RG 

can be employed to derive the asymptotic scaling laws in stochastic dynamical 

systems; to infer the upper critical dimension d c (for dimensions d ≤ d c , 

fluctuations strongly affect the universal scaling properties); and to systematically 

compute the critical exponents as well as to determine further universal 

properties in various intriguing dynamical model systems both near and far 

from equilibrium. (For considerably more details, especially on the more technical 

aspects, the reader is referred to Ref. [13].) 

7.1.1 Continuous Phase Transitions and Critical Slowing Down 

The vicinity of a critical point is characterised by strong correlations and large 

fluctuations. The system under investigation is then behaving in a highly cooperative 

manner, and as a consequence, the standard approximative methods of 

statistical mechanics, namely perturbation or cluster expansions that assume 

either weak interactions or short-range correlations, fail. Upon approaching 

an equilibrium continuous (second-order) phase transition, i.e., for |τ| ≪1, 

where τ =(T − T c )/T c measures the deviation from the critical temperature 

T c , the thermal fluctuations of the order parameter S(x) (which characterises 

the different thermodynamic phases, usually chosen such that the thermal 

average 〈S〉 = 0 vanishes in the high-temperature “disordered” phase) are, in 

the thermodynamic limit, governed by a diverging length scale 

ξ(τ) ∼|τ| −ν . (7.1) 

Here, we have defined the correlation length via the typically exponential 

decay of the static cumulant or connected two-point correlation function 

C(x) =〈S(x) S(0)〉 −〈S〉 2 ∼ e −|x|/ξ ,andν denotes the correlation length 

critical exponent. AsT → T c , ξ →∞, which entails the absence of any characteristic 

length scale for the order parameter fluctuations at criticality. Hence 

we expect the critical correlations to follow a power law C(x) ∼|x| −(d−2+η) 

in d dimensions, which defines the Fisher exponent η. The following scaling 

ansatz generalises this power law to T ≠ T c , but still in the vicinity of the 

critical point, 

C(τ,x) =|x| −(d−2+η) ˜C± (x/ξ) , (7.2) 

with two distinct regular scaling functions ˜C + (y) forT>T c and ˜C − (y) for 

T

7 Field-Theory Approaches to Nonequilibrium Dynamics 297 

As we will see in Subsect. 7.1.5, there are only two independent static 

critical exponents. Consequently, it must be possible to use the static scaling 

hypothesis (7.2) or (7.3), along with the definition (7.1), to express the 

exponents describing the thermodynamic singularities near a second-order 

phase transition in terms of ν and η through scaling laws. For example, the 

order parameter in the low-temperature phase (τ < 0) is expected to grow 

as 〈S〉 ∼ (−τ) β . Let us consider Eq. (7.2) in the limit |x| → ∞.Inorder 

for the |x| dependence to cancel, ˜C± (y) ∝|y| d−2+η for large |y|, and 

therefore C(τ,|x| → ∞) ∼ ξ −(d−2+η) ∼ |τ| ν(d−2+η) . On the other hand, 

C(τ,|x| →∞) →−〈S〉 2 ∼−(−τ) 2β for T < T c ; thus we identify the order 

parameter critical exponent through the hyperscaling relation 

β = ν (d − 2+η) . (7.4) 

2 

Let us next consider the isothermal static susceptibility χ τ , which according to 

the equilibrium fluctuation–response theorem is given in terms of the spatial 

integral of the correlation function C(τ,x): χ τ (τ) =(k B T ) −1 lim q→0 C(τ,q). 

But Ĉ±(p) ∼ |p| 2−η as p → 0 to ensure nonsingular behaviour, whence 

χ τ (τ) ∼ ξ 2−η ∼|τ| −ν(2−η) , and upon defining the associated thermodynamic 

critical exponent γ via χ τ (τ) ∼|τ| −γ , we obtain the scaling relation 

γ = ν (2 − η) . (7.5) 

The scaling laws (7.2), (7.3) as well as scaling relations such as (7.4) and 

(7.5) can be put on solid foundations by means of the RG procedure, based on 

an effective long-wavelength Hamiltonian H[S], a functional of S(x), that captures 

the essential physics of the problem, namely the relevant symmetries in 

order parameter and real space, and the existence of a continuous phase transition. 

The probability of finding a configuration S(x) at given temperature 

T is then given by the canonical distribution 

P eq [S] ∝ exp (−H[S]/k B T ) . (7.6) 

For example, the mathematical description of the critical phenomena for an 

O(n)-symmetric order parameter field S α (x), with vector index α =1,...,n, 

isbasedontheLandau–Ginzburg–Wilson functional [1–6] 

∫ 

H[S] = d d x ∑ [ r 

2 [Sα (x)] 2 + 1 2 [∇Sα (x)] 2 

α 

+ u 4! [Sα (x)] ∑ ] 

2 [S β (x)] 2 − h α (x) S α (x) , (7.7) 

β 

where h α (x) is the external field thermodynamically conjugate to S α (x), 

u>0 denotes the strength of the nonlinearity that drives the phase transformation, 

and r is the control parameter for the transition, i.e., r ∝ T − T 0 c ,


where Tc 

0 is the (mean-field) critical temperature. Spatial variations of the 

order parameter are energetically suppressed by the term ∼ [∇S α (x)] 2 ,and 

the corresponding positive coefficient has been absorbed into the fields S α . 

We shall, however, not pursue the static theory further here, but instead 

proceed to a full dynamical description in terms of nonlinear Langevin equations 

[7, 8]. We will then formulate the RG within this dynamic framework, 

and therein demonstrate the emergence of scaling laws and the computation 

of critical exponents in a systematic perturbative expansion with respect to 

the deviation ɛ = d − d c from the upper critical dimension. 

In order to construct the desired effective stochastic dynamics near a critical 

point, we recall that correlated region of size ξ become quite large in the 

vicinity of the transition. Since the associated relaxation times for such clusters 

should grow with their extent, one would expect the characteristic time 

scale for the relaxation of the order parameter fluctuations to increase as well 

as T → T c , namely 

t c (τ) ∼ ξ(τ) z ∼|τ| −zν , (7.8) 

which introduces the dynamic critical exponent z that encodes the critical 

slowing down at the phase transition; usually z ≥ 1. Since the typical relaxation 

rates therefore scale as ω c (τ) =1/t c (τ) ∼|τ| zν , we may utilise the 

static scaling variable p = q ξ to generalise the crucial observation (7.8) and 

formulate a dynamic scaling hypothesis for the wavevector-dependent dispersion 

relation of the order parameter fluctuations [14, 15], 

ω c (τ,q) =|q| z ˆω ± (q ξ) . (7.9) 

We can then proceed to write down dynamical scaling laws by simply 

postulating the additional scaling variables s = t/t c (τ) orω/ω c (τ,q). For 

example, as an immediate consequence we find for the time-dependent mean 

order parameter 

〈S(τ,t)〉 = |τ| β Ŝ(t/t c ) , (7.10) 

with Ŝ(s →∞) = const., but Ŝ(s) ∼ s−β/zν as s → 0 in order for the τ 

dependence to disappear. At the critical point (τ = 0), this yields the powerlaw 

decay 〈S(t)〉 ∼t −α , with 

α = β 

zν = 1 (d − 2+η) . (7.11) 

2 z 

Similarly, the scaling law for the dynamic order parameter susceptibility (response 

function) becomes 

χ(τ,q,ω)=|q| −2+η ˆχ ± (q ξ,ω ξ z ) , (7.12) 

which constitutes the dynamical generalisation of Eq. (7.3), for χ(τ,q, 0) = 

(k B T ) −1 C(τ,q). Upon applying the fluctuation–dissipation theorem, valid in 

thermal equilibrium, we therefrom obtain the dynamic correlation function

C(τ,q,ω)= 2k BT 

ω 


Im χ(τ,q,ω)=|q| −z−2+η Ĉ ± (q ξ,ω ξ z ) , (7.13) 

and for its Fourier transform in real space and time, 

∫ d d ∫ 

q dω 

C(τ,x,t)= 

ei(q·x−ωt) (2π) d C(τ,q,ω)=|x| −(d−2+η) ˜C± (x/ξ, t/ξ z ) , 

2π 

(7.14) 

which reduces to the static limit (7.2) if we set t =0. 

The critical slowing down of the order parameter fluctuations near the 

critical point provides us with a natural separation of time scales. Assuming 

(for now) that there are no other conserved variables in the system, which 

would constitute additional slow modes, we may thus resort to a coarse-grained 

long-wavelength and long-time description, focusing merely on the order parameter 

kinetics, while subsuming all other “fast” degrees of freedom in random 

“noise” terms. This leads us to a mesoscopic Langevin equation for the slow 

variables S α (x,t)oftheform 

∂S α (x,t) 

= F α [S](x,t)+ζ α (x,t) . (7.15) 

∂t 

In the simplest case, the systematic force terms here just represent purely 

relaxational dynamics towards the equilibrium configuration [16], 

F α [S](x,t)=−D 

δH[S] 

δS α (x, t) , (7.16) 

where D represents the relaxation coefficient, and H[S] is again the effective 

Hamiltonian that governs the phase transition, e.g. given by Eq. (7.7). For 

the stochastic forces we may assume the most convenient form, and take them 

to simply represent Gaussian white noise with zero mean, 〈ζ α (x,t)〉 = 0, but 

with their second moment in thermal equilibrium fixed by Einstein’s relation 

〈 

ζ α (x,t) ζ β (x ′ ,t ′ ) 〉 =2k B TDδ(x − x ′ ) δ(t − t ′ ) δ αβ . (7.17) 

As can be verified by means of the associated Fokker–Planck equation for the 

time-dependent probability distribution P[S, t], Eq. (7.17) guarantees that 

eventually P[S, t →∞] →P eq [S], the canonical distribution (7.6). The stochastic 

differential equation (7.15), with (7.16), the Hamiltonian (7.7), and 

the noise correlator (7.17), define the relaxational model A (according to the 

classification in Ref. [7]) for a nonconserved O(n)-symmetric order parameter. 

If, however, the order parameter is conserved, wehavetoconsidertheassociated 

continuity equation ∂ t S α +∇·J α = 0, where typically the conserved 

current is given by a gradient of the field S α : J α = −D ∇S α + ...; as a consequence, 

the order parameter fluctuations will relax diffusively with diffusion 

coefficient D. The ensuing model B [7,16] for the relaxational critical dynamics 

of a conserved order parameter can be obtained by replacing D →−D ∇ 2 in 

Eqs. (7.16) and (7.17). In fact, we will henceforth treat both models A and B


simultaneously by setting D → D (i∇) a , where a =0anda = 2 respectively 

represent the nonconserved and conserved cases. Explicitly, we thus obtain 

∂S α (x,t) 

∂t 

= −D (i∇) a δH[S] 

δS α (x,t) + ζα (x,t) 

= −D (i∇) a[ r − ∇ 2 + u ∑ 

[S β (x)] 2] S α (x,t) 

6 

β 

+D (i∇) a h α (x,t)+ζ α (x,t) , (7.18) 

with 

〈 

ζ α (x,t) ζ β (x ′ ,t ′ ) 〉 =2k B TD(i∇) a δ(x − x ′ ) δ(t − t ′ ) δ αβ . (7.19) 

Notice already that the presence or absence of a conservation law for the order 

parameter implies different dynamics for systems described by identical static 

behaviour. Before proceeding with the analysis of the relaxational models, 

we remark that in general there may exist additional reversible contributions 

to the systematic forces F α [S], see Subsect. 7.1.6, and/or dynamical modecouplings 

to additional conserved, slow fields, which effect further splitting 

into several distinct dynamic universality classes [6, 7, 13]. 

Let us now evaluate the dynamic response and correlation functions in the 

Gaussian (mean-field) approximation in the high-temperature phase. To this 

end, we set u = 0 and thus discard the nonlinear terms in the Hamiltonian 

(7.7) as well as in Eq. (7.18). The ensuing Langevin equation becomes linear in 

the fields S α , and is therefore readily solved by means of Fourier transforms. 

Straightforward algebra and regrouping some terms yields 

[ 

−iω + Dq 

a ( r + q 2)] S α (q,ω)=Dq a h α (q,ω)+ζ α (q,ω) . (7.20) 

With 〈ζ α (q,ω)〉 = 0, this gives immediately 

χ αβ 

(q,ω)〉 

0 (q,ω)=∂〈Sα ∂h β (q,ω) ∣ = Dq a G 0 (q,ω) δ αβ , (7.21) 

h=0 

with the response propagator 

G 0 (q,ω)= [ −iω + Dq a (r + q 2 ) ] −1 

. (7.22) 

As is readily established by means of the residue theorem, its Fourier backtransform 

in time obeys causality, 

G 0 (q,t)=Θ(t)e −Dqa (r+q 2 ) t . (7.23) 

Setting h α = 0, and with the noise correlator (7.19) in Fourier space 

〈 

ζ α (q,ω) ζ β (q ′ ,ω ′ ) 〉 =2k B TDq a (2π) d+1 δ(q + q ′ ) δ(ω + ω ′ ) δ αβ , (7.24)


we obtain the Gaussian dynamic correlation function 〈 S α (q,ω) S β (q ′ ,ω ′ ) 〉 0 = 

C 0 (q,ω)(2π) d+1 δ(q + q ′ ) δ(ω + ω ′ ), where 

C 0 (q,ω)= 

2k B TDq a 

ω 2 +[Dq a (r + q 2 )] 2 =2k BTDq a |G 0 (q,ω)| 2 . (7.25) 

The fluctuation–dissipation theorem (7.13) is of course satisfied; moreover, as 

function of wavevector and time, 

C 0 (q,t)= 

k BT 

r + q 2 e−Dqa (r+q 2 ) |t| . (7.26) 

In the Gaussian approximation, away from criticality (r > 0, q ≠ 0) the 

temporal correlations for models A and B decay exponentially, with the relaxation 

rate ω c (r, q) = Dq 2+a (1 + r/q 2 ). Upon comparison with the dynamic 

scaling hypothesis (7.9), we infer the mean-field scaling exponents 

ν 0 =1/2 andz 0 =2+a. At the critical point, a nonconserved order parameter 

relaxes diffusively (z 0 = 2) in this approximation, whereas the conserved 

order parameter kinetics becomes even slower, namely subdiffusive with 

z 0 = 4. Finally, invoking Eqs. (7.12), (7.13), (7.14), or simply the static limit 

C 0 (q, 0) = k B T/(r + q 2 ), we find η 0 = 0 for the Gaussian model. 

The full nonlinear Langevin equation (7.18) cannot be solved exactly. Yet 

a perturbation expansion with respect to the coupling u may be set up in a 

slightly cumbersome, but straightforward manner by direct iteration of the 

equations of motion [16, 17]. More elegantly, one may utilise a path-integral 

representation of the Langevin stochastic process [18, 19], which allows the 

application of all the standard tools from statistical and quantum field theory 

[1–6], and has the additional advantage of rendering symmetries in the 

problem more explicit [8, 9, 13]. 

7.1.2 Field Theory Representation of Langevin Equations 

Our starting point is a set of coupled Langevin equations of the form (7.15) 

for mesoscopic, coarse-grained stochastic variables S α (x,t). For the stochastic 

forces, we make the simplest possible assumption of Gaussian white noise, 

〈ζ α (x,t)〉 =0, 

〈 

ζ α (x,t) ζ β (x ′ ,t ′ ) 〉 =2L α δ(x − x ′ ) δ(t − t ′ ) δ αβ , (7.27) 

where L α may represent a differential operator (such as the Laplacian ∇ 2 for 

conserved fields), and even a functional of S α . In the time interval 0 ≤ t ≤ t f , 

the moments (7.27) are encoded in the probability distribution 

[ 

W[ζ] ∝ exp − 1 ∫ ∫ tf 

d d x dt ∑ ζ α (x,t) [ (L α ) −1 ζ α (x,t) ]] . (7.28) 

4 0 α 

If we now switch variables from the stochastic noise ζ α to the fields S α by 

means of the equations of motion (7.15), we obtain


W[ζ] D[ζ] =P[S] D[S] ∝ e −G[S] D[S] , (7.29) 

with the statistical weight determined by the Onsager–Machlup functional [9] 

G[S] = 1 ∫ ∫ 

d d x dt ∑ ( ( )] 

∂S 

α 

∂S 

− F α [S] 

)[(L α ) −1 α 

− F α [S] . 

4 

∂t 

∂t 

α 

(7.30) 

Note that the Jacobian for the nonlinear variable transformation {ζ α }→ 

{S α } has been omitted here. In fact, the above procedure is properly defined 

through appropriately discretising time. If a forward (Itô) discretisation is applied, 

then indeed the associated functional determinant is a mere constant 

that can be absorbed in the functional measure. The functional (7.30) already 

represents a desired field theory action. Since the probability distribution for 

the stochastic forces should be normalised, ∫ D[ζ] W [ζ] = 1, the associated 

“partition function” is unity, and carries no physical information (as opposed 

to static statistical field theory, where it determines the free energy and hence 

the entire thermodynamics). The Onsager–Machlup representation is however 

plagued by technical problems: Eq. (7.30) contains (L α ) −1 , which for 

conserved variables entails the inverse Laplacian operator, i.e., a Green function 

in real space or the singular factor 1/q 2 in Fourier space; moreover the 

nonlinearities in F α [S] appear quadratically. Hence it is desirable to linearise 

the action (7.30) by means of a Hubbard–Stratonovich transformation [9]. 

We shall follow an alternative, more general route that completely avoids 

the appearance of the inverse operators (L α ) −1 in intermediate steps. Our 

goal is to average over noise “histories” for observables A[S] that need to be 

expressible in terms of the stochastic fields S α : 〈A[S]〉 ζ ∝ ∫ D[ζ] A[S(ζ)] W [ζ]. 

For this purpose, we employ the identity 

∫ 

1= D[S] ∏ ∏ 

( ∂S α ) 

(x,t) 

δ 

− F α [S](x,t) − ζ α (x,t) 

∂t 

α (x,t) 

∫ ∫ [ ∫ ∫ 

= D[i ˜S] D[S]exp − d d x dt ∑ ( )] 

∂S 

˜S α α 

− F α [S] − ζ α , (7.31) 

∂t 

α 

where the first line constitutes a rather involved representation of the unity (in 

a somewhat symbolic notation; again proper discretisation should be invoked 

here), and the second line utilises the Fourier representation of the (functional) 

delta distribution by means of the purely imaginary auxiliary fields ˜S (and 

factors 2π have been absorbed in its functional measure). 

Inserting (7.31) and the probability distribution (7.28) into the desired 

stochastic noise average, we arrive at 

∫ ∫ [ ∫ ∫ 

〈A[S]〉 ζ ∝ D[i ˜S] D[S]exp − d d x dt ∑ ( )] 

∂S 

˜S α α 

− F α [S] A[S] 

∂t 

α 

∫ ( ∫ ∫ 

× D[ζ]exp − d d x dt ∑ [ ]) 

1 

4 ζα (L α ) −1 ζ α − ˜S α ζ α . (7.32) 

α


We may now evaluate the Gaussian integrals over the noise ζ α , which yields 

∫ 

∫ 

〈A[S]〉 ζ = D[S] A[S] P[S] , P[S] ∝ D[i ˜S]e −A[ ˜S,S] , (7.33) 

with the statistical weight now governed by the Janssen–De Dominicis “response” 

functional [9, 18, 19] 

∫ ∫ 

A[ ˜S,S]= 

tf 

d d x dt ∑ [ ( ) 

] 

∂S 

˜S α α 

− F α [S] − 

0 

∂t 

˜S α L α ˜Sα . (7.34) 

α 

Once again, we have omitted the functional determinant from the variable 

change {ζ α }→{S α }, and normalisation implies ∫ D[i ˜S] ∫ D[S]e −A[ ˜S,S] =1. 

The first term in the action (7.34) encodes the temporal evolution according 

to the systematic terms in the Langevin equations (7.15), whereas the second 

term specifies the noise correlations (7.27). Since the auxiliary variables ˜S α , 

often termed Martin–Siggia–Rose response fields [20], appear only quadratically 

here, they may be eliminated via completing the squares and Gaussian 

integrations; thereby one recovers the Onsager–Machlup functional (7.30). 

The Janssen–De Dominicis functional (7.34) takes the form of a (d + 1)- 

dimensional statistical field theory with two independent sets of fields S α and 

˜S α . We may thus bring the established machinery of statistical and quantum 

field theory [1–6] to bear here; it should however be noted that the response 

functional formalism for stochastic Langevin dynamics incorporates causality 

in a nontrivial manner, which leads to important distinctions [8]. 

Let us specify the Janssen–De Dominicis functional for the purely relaxational 

models A and B [16, 17], see Eqs. (7.18) and (7.19), splitting it into 

the Gaussian and anharmonic parts A = A 0 + A int [9], which read 

∫ ∫ 

A 0 [ ˜S,S] = d d x dt ∑ ( [ ] 

∂ 

˜S α ∂t + D (i∇)a (r − ∇ 2 ) S α 

α 

) 

−D ˜S α (i∇) a ˜Sα − D ˜S α (i∇) a h α , (7.35) 

A int [ ˜S,S] =D u ∫ ∫ 

d d x dt ∑ ˜S α (i∇) a S α S β S β . (7.36) 

6 

α,β 

Since we are interested in the vicinity of the critical point T ≈ T c ,wehave 

absorbed the constant k B T c into the fields. The prescription (7.33) tells us how 

to compute time-dependent correlation functions 〈 S α (x,t) S β (x ′ ,t ′ ) 〉 .Using 

Eq. (7.35), the dynamic order parameter susceptibility follows from 

χ αβ (x − x ′ ,t− t ′ )= δ〈Sα (x,t)〉 

δh β (x ′ ,t ′ ) 

∣ = D 

h=0 

〈 

〉 

S α (x,t)(i∇) a ˜Sβ (x ′ ,t ′ ) 

(7.37) 

for the simple relaxational models (only), the response function is just given by 

a correlator that involves an auxiliary variable, which explains why the ˜S α are 

;


referred to as “response” fields. In equilibrium, one may employ the Onsager– 

Machlup functional (7.30) to derive the fluctuation–dissipation theorem [9] 

χ αβ (x − x ′ ,t− t ′ )=Θ(t − t ′ ) ∂ 

∂t ′ 〈 

S α (x,t) S β (x ′ ,t ′ ) 〉 , (7.38) 

which is equivalent to Eq. (7.13) in Fourier space. 

In order to access arbitrary correlators, we define the generating functional 

〈 ∫ ∫ 

Z[˜j,j]= exp d d x dt ∑ (˜j α ˜Sα + j α S α)〉 , (7.39) 

α 

wherefrom the correlation functions follow via functional derivatives, 

〈 ∏ 

〉 

S αi ˜S αj = ∏ δ δ 

Z[˜j,j] ∣ , (7.40) 

δj αi δ˜j 

ij 

ij 

αj ∣˜j=0=j 

and the cumulants or connected correlation functions via 

〈 ∏ 

〉 

S αi ˜S αj = ∏ δ δ 

ln Z[˜j,j] ∣ 

ij 

c 

δj αi δ˜j 

ij 

αj 

∣˜j=0=j 

. (7.41) 

In the harmonic approximation, setting u =0,Z[˜j,j] can be evaluated explicitly 

(most directly in Fourier space) by means of Gaussian integration [9,13]; 

one thereby recovers (with k B T = 1) the Gaussian response propagator (7.22) 

and two-point 〈 correlation function (7.25). Moreover, as a consequence of 

causality, ˜Sα (q,ω) ˜S β (q ′ ,ω )〉 

′ =0. 

7.1.3 Outline of Dynamic Perturbation Theory 

0 

Since we cannot evaluate correlation functions with the nonlinear action (7.36) 

exactly, we resort to a perturbational treatment, assuming, for the time being, 

a small coupling strength u. Theperturbation expansion with respect to u is 

constructed by rewriting the desired correlation functions in terms of averages 

with respect to the Gaussian action (7.35), henceforth indicated with index 

’0’, and then expanding the exponential of −A int , 

〈 ∏ 

ij 

S αi ˜S αj 〉 

= 

〈 ∏ ˜S ij Sαi αj −Aint[ ˜S,S]〉 

e 

〈 ∏ 

= 

ij 

〈〉 

e−Aint[ ˜S,S] 

0 

S αi ˜S ∑ ∞ αj 

l=0 

1 

l! 

0 

( 

−A int [ ˜S,S] 

) l 

〉 

0 

. (7.42) 

The remaining Gaussian averages, a series of polynomials in the fields S α 

and ˜S α , can be evaluated by means of Wick’s theorem, here an immediate


consequence of the Gaussian statistical weight, which states that all such averages 

can be written as a sum over all possible factorisations into Gaussian 

two-point functions 〈S α ˜Sβ 〉 0 , i.e., essentially the response propagator G 0 , 

Eq. (7.22), and 〈S α S β 〉 0 , the Gaussian correlation function C 0 , Eq. (7.25). 

Recall that the denominator in Eq. (7.42) is exactly unity as a consequence of 

normalisation; alternatively, this result follows from causality in conjunction 

with our forward descretisation prescription, which implies that we should 

identify Θ(0) = 0. (We remark that had we chosen another temporal discretisation 

rule, any apparent contributions from the denominator would be 

precisely cancelled by the in this case nonvanishing functional Jacobian from 

the variable transformation {ζ α }→{S α }.) At any rate, our stochastic field 

theory contains no “vacuum” contributions. 

The many terms in the perturbation expansion (7.42) are most lucidly 

organised in a graphical representation, using Feynman diagrams with the 

basic elements depicted in Fig. 7.1. We represent the response propagator 

(7.22) by a directed line (here conventionally from right to left), which encodes 

its causal nature; the noise by a two-point “source” vertex, and the anharmonic 

term in Eq. (7.36) as a four-point vertex. In the diagrams representing the 

different terms in the perturbation series, these vertices serve as links for the 

propagator lines, with the fields S α being encoded as the “incoming”, and the 

˜S α as the “outgoing” components of the lines. In Fourier space, translational 

invariance in space and time implies wavevector and frequency conservation 

at each vertex, see Fig. 7.2 below. An alternative, equivalent representation 

uses both the response and correlation propagators as independent elements, 

the latter depicted as undirected line, thereby disposing of the noise vertex, 

and retaining the nonlinearity in Fig. 7.1(c) as sole vertex. 

Following standard field theory procedures [1–5], one establishes that the 

perturbation series for the cumulants (7.41) is given in terms of connected 

Feynman graphs only (for a detailed exposition of this and the following results, 

see Ref. [13]). An additional helpful reduction in the number of diagrams 

to be considered arises when one considers the vertex functions, which generalise 

the self-energy contributions Σ(q,ω)intheDyson equation for the 

response propagator, G(q,ω) −1 = Dq a χ(q,ω) −1 = G 0 (q,ω) −1 − Σ(q,ω). 

(a) 

α 

q,ω 

β 

= 

1 

i ω+ D q 

a 

( r + q2) 

δ αβ 

(b) 

α 

β 

q 

-q 

= 

2 D q a δ αβ 

(c) 

α 

q 

α 

β 

β 

= 

D q a 

u 

6 

Fig. 7.1. Elements of dynamic perturbation theory for the O(n)-symmetric relaxational 

models: (a) response propagator; (b) noisevertex;(c) anharmonic vertex


To this end, we define the fields ˜Φ α = δ ln Z/δ˜j α and Φ α = δ ln Z/δj α ,and 

introduce the new generating functional 

∫ ∫ 

Γ [ ˜Φ, Φ] =− ln Z[˜j,j]+ d d x dt ∑ (˜j α ˜Φα + j α Φ α) , (7.43) 

α 

wherefrom the vertex functions are obtained via the functional derivatives 

Γ (Ñ,N) 

{α i};{α j} = Ñ 

∏ 

i 

N 

δ 

αi 

δ ˜Φ 

∏ 

δ 

δΦ Γ [ ˜Φ, Φ] ∣ 

αj 

j 

∣˜j=0=j 

. (7.44) 

Diagrammatically, these quantities turn out to be represented by the possible 

sets of one-particle (1PI) irreducible Feynman graphs with N incoming and 

Ñ outgoing “amputated” legs; i.e., these diagrams do not split into allowed 

subgraphs by simply cutting any single propagator line. For example, for the 

two-point functions a direct calculation yields the relations 

Γ (1,1) (q,ω)=Dq a χ(−q, −ω) −1 = G 0 (−q, −ω) −1 − Σ(−q, −ω) ,(7.45) 

Γ (2,0) (q,ω)=− 

C(q,ω) qa 

= −2D Im Γ (1,1) (q,ω) , (7.46) 

|G(q,ω)| 

2 

ω 

where the second equation for Γ (2,0) follows from the fluctuation–dissipation 

theorem (7.13). Note that Γ (0,2) (q,ω) = 0 vanishes because of causality. 

The perturbation series can then be organised graphically as an expansion 

in successive orders with respect to the number of closed propagator loops. 

As an example, Fig. 7.2 depicts the one-loop contributions for the vertex 

functions Γ (1,1) and Γ (1,3) in the time domain with all required labels. One 

may formulate general Feynman rules for the construction of the diagrams and 

their translation into mathematical expressions for the lth order contribution 

to the vertex function Γ (Ñ,N) : 

1. Draw all topologically different, connected one-particle irreducible graphs 

with Ñ outgoing and N incoming lines connecting l relaxation vertices ∝ 

u. Donot allow closed response loops (since in the Itô calculus Θ(0) = 0). 

2. Attach wavevectors q i , frequencies ω i or times t i , and component indices 

α i to all directed lines, obeying “momentum (and energy)” conservation 

at each vertex. 

(a) 

α 

k 

t 

t´ 

β -k 

α 

(b) 

α 

0 

q-k 

β 

t 

k 

α 

α 

α 

-k 

t´ 

Fig. 7.2. One-loop diagrams for (a) Γ (1,1) and (b) Γ (1,3) in the time domain


3. Each directed line corresponds to a response propagator G 0 (−q, −ω) or 

G 0 (q,t i −t j ) in the frequency and time domain, respectively, the two-point 

vertex to the noise strength 2D q a , and the four-point relaxation vertex 

to −D q a u/6. Closed loops imply integrals over the internal wavevectors 

and frequencies or times, subject to causality constraints, as well as sums 

over the internal vector indices. Apply the residue theorem to evaluate 

frequency integrals. 

4. Multiply with −1 and the combinatorial factor counting all possible ways 

of connecting the propagators, l relaxation vertices, and k two-point vertices 

leading to topologically identical graphs, including a factor 1/l! k! 

originating in the expansion of exp(−A int [ ˜S,S]). 

For later use, we provide the explicit results for the two-point vertex functions 

to two-loop order. After some algebra, the three diagrams in Fig. 7.3 

give 

[ 

Γ (1,1) (q,ω)=iω + Dq a r + q 2 + n +2 

6 

( ) 2 ∫ 

n +2 

− u 

6 

k 

− n +2 

× 

18 u2 ∫k 

( 

1 − 

∫ 

1 

r + k 2 

∫ 

1 

r + k 2 

∫ 

u 

k 

k ′ 1 

1 

r + k 2 

1 

k ′ (r + k ′2 ) 2 

1 

r + k ′ 2 

r +(q − k − k ′ ) 

)] 

2 

, (7.47) 

iω 

iω + ∆(k)+∆(k ′ )+∆(q − k − k ′ ) 

where we have separated out the dynamic part in the last line, and introduced 

the abbreviations ∆(q) =Dq a (r + q 2 )and ∫ k = ∫ d d k/(2π) d [13]. For the 

noise vertex, Fig. 7.4(a) yields [13] 

[ 

Γ (2,0) (q,ω)=−2Dq a 1+Dq a n +2 

18 

∫k 

u2 

1 

× 

r +(q − k − k ′ ) 2 Re 1 

∫ 

1 

r + k 2 

k ′ 1 

r + k ′ 2 

] 

iω + ∆(k)+∆(k ′ )+∆(q − k − k ′ ; (7.48) 

) 

notice that for model B, as a consequence of the conservation law for the 

order parameter and ensuing wavevector dependence of the nonlinear vertex, 

+ + + ... 

Fig. 7.3. One-particle irreducible diagrams for Γ (1,1) (q,ω) to second order in u


(a) 

(b) 

Fig. 7.4. (a) Two-loop diagram for Γ (2,0) (q,ω); (b) one-loop graph for Γ (1,3) 

see Fig. 7.1(c), to all orders in the perturbation expansion 

a =2 : Γ (1,1) ∂ 

(q =0,ω)=iω, 

∂q 2 Γ (2,0) (q,ω) 

∣ = −2D . (7.49) 

q=0 

At last, with the shorthand notation k =(q,ω), the analytical expression 

corresponding to the graph in Fig. 7.4(b) for the four-point vertex function 

at symmetrically chosen external wavevector labels reads 

( ) a [ 3 

Γ (1,3) (−3k/2; {k/2}) =D 

2 q u 

∫ 

( 

1 1 

× 

k r + k 2 r +(q − k) 2 1 − 

7.1.4 Renormalisation 

1 − n +8 

6 

u 

iω 

iω + ∆(k)+∆(q − k) 

)] 

. (7.50) 

Consider a typical loop integral, say the correction in Eq. (7.50) to the fourpoint 

vertex function Γ (1,3) at zero external frequency and momentum, whose 

“bare” value, without any fluctuation contributions, is u. In dimensions d


Conversely, for dimensions d ≥ 4, the integral in (7.51) develops ultraviolet 

(UV) divergences as the upper integral boundary is sent to infinity (k = |k|), 

∫ Λ 

k d−1 { } 

ln(Λ 

(r + k 2 ) 2 dk ∼ 2 /r) d =4 

Λ d−4 →∞ as Λ →∞. (7.52) 

d>4 

0 

In lattice models, there is a finite wavevector cutoff, namely the Brillouin 

zone boundary, Λ ∼ (2π/a 0 ) d for a hypercubic lattice with lattice constant 

a 0 , whence physically these UV problems do not emerge. Yet we shall see that 

a formal treatment of these unphysical UV divergences will allow us to infer 

the correct power laws for the physical IR singularities associated with the 

critical point. The borderline dimension that separates the IR and UV singular 

regimes is referred to as upper critical dimension d c ; here d c = 4. Note that 

at d c , UV and IR singularities are intimately connected and appear in the 

form of logarithmic divergences, see Eq. (7.52). The situation is summarised 

in Table 7.1, where we have also stated that models with continuous order 

parameter symmetry, such as the Hamiltonian (7.7) with n ≥ 2, do not allow 

long-range order in dimensions d ≤ d lc = 2 (Mermin–Wagner–Hohenberg 

theorem [21–23]). Here, d lc is called the lower critical dimension; for the Ising 

model represented by Eq. (7.7) with n =1,ofcoursed lc =1. 

Table 7.1. Mathematical and physical distinctions of the regimes dd c, for the O(n)-symmetric models A and B (or static Φ 4 field theory) 

Dimension Perturbation Model A / B or Critical 

Interval Series Φ 4 Field Theory Behaviour 

d ≤ d lc = 2 IR-singular ill-defined no long-range 

UV-convergent u relevant order (n ≥ 2) 

2


phase transition; but u becomes irrelevant for d>4: one then expects meanfield 

(Gaussian) critical exponents. At the upper critical dimension d c =4, 

the nonlinear coupling u is marginally relevant: this will induce logarithmic 

corrections to the mean-field scaling laws, see Table 7.1. 

It is obviously not a simple task to treat the IR-singular perturbation 

expansion in a meaningful, well-defined manner, and thus allow nonanalytic 

modifications of the critical power laws (note that mean-field scaling is completely 

determined by dimensional analysis or power counting). The key of 

the success of the RG approach is to focus on the very specific symmetry 

that emerges near critical points, namely scale invariance. There are several 

(largely equivalent) versions of the RG method; we shall here formulate 

and employ the field-theoretic variant [1–6, 8, 13]. In order to proceed, 

it is convenient to evaluate the loop integrals in momentum space by means 

of dimensional regularisation, whereby one assigns finite values even to UVdivergent 

expressions, namely the analytically continued values from the UVfinite 

range. For example, even for noninteger dimensions d and σ, weset 

∫ 

d d k 

(2π) d 

k 2σ Γ (σ + d/2) Γ (s − σ − d/2) 

(τ + k 2 ) s = 

2 d π d/2 Γ (d/2) Γ (s) 

The renormalisation program then consists of the following steps: 

τ σ−s+d/2 . (7.53) 

1. We aim to carefully keep track of formal, unphysical UV divergences. In 

dimensionally regularised integrals (7.53), these appear as poles in ɛ = 

d c − d; their residues characterise the asymptotic UV behaviour of the field 

theory under consideration. 

2. Therefrom we may infer the (UV) scaling properties of the control parameters 

of the model under a RG transformation, namely essentially a change 

of the momentum scale µ, while keeping the form of the action invariant. 

This will allow us to define suitable running couplings. 

3. We seek fixed points in parameter space where certain marginal couplings 

(u here) do not change anymore under RG transformations. This describes 

a scale-invariant regime for the model under consideration, where the UV 

and IR scaling properties become intimately linked. Studying the parameter 

flows near a stable RG fixed point then allows us to extract the 

asymptotic IR power laws. 

As a preliminary step, we need to take into account that the fluctuations 

will also shift the critical point downwards from the mean-field phase transition 

temperature T 0 c ; i.e., we expect the transition to occur at T c


Notice that this quantity depends on microscopic details (the lattice structure 

enters the cutoff Λ) and is thus not universal; moreover it diverges for d ≥ 2 

(quadratically near d c =4)asΛ →∞.Wenextuser = τ + r c to write 

physical quantities as functions of the true distance τ from the critical point, 

which technically amounts to an additive renormalisation; e.g., the dynamic 

response function becomes to one-loop order 

χ(q,ω) −1 = − iω 

Dq a + q2 + τ 

[ 

1 − n +2 u 

6 

∫ 

k 

] 

1 

k 2 (τ + k 2 + O(u 2 ) . (7.55) 

) 

The remaining loop integral is UV-singular in dimensions d ≥ d c =4. 

We may now formally absorb the remaining UV divergences into renormalised 

fields and parameters, a procedure called multiplicative renormalisation. 

For the renormalised fields, we use the convention 

SR α = Z 1/2 

S 

S α , ˜Sα R = Z 1/2 ˜S α , (7.56) 

where we have exploited the O(n) rotational symmetry in using identical 

renormalisation constants (Z factors) for each component. The renormalised 

cumulants with N order parameter fields S α and Ñ response fields ˜S α naturally 

involve the product Z N/2 

S 

Γ (Ñ,N) 

R 

ZÑ/2 , whence 

˜S 

˜S 

= Z −Ñ/2 Z −N/2 

˜S S 

Γ (Ñ,N) . (7.57) 

In a similar manner, we relate the “bare” parameters of the theory via Z 

factors to their renormalised counterparts, which we furthermore render dimensionless 

through appropriate momentum scale factors, 

D R = Z D D, τ R = Z τ τµ −2 , u R = Z u uA d µ d−4 , (7.58) 

where we have separated out the factor A d = Γ (3 − d/2)/2 d−1 π d/2 for convenience. 

In the minimal subtraction scheme, the Z factors contain only the 

UV-singular terms, which in dimensional regularisation appear as poles at 

ɛ = 0, and their residues, evaluated at d = d c . 

These renormalisation constants are not all independent, however; since 

the equilibrium fluctuation–dissipation theorem (7.38) or (7.13) must hold in 

the renormalised theory as well, we infer that necessarily 

and consequently from Eq. (7.45) 

Z D = ( Z S /Z ˜S) 1/2 

, (7.59) 

χ R = Z S χ. (7.60) 

Moreover, for model B with conserved order parameter Eq. (7.49) implies that 

to all orders in the perturbation expansion


a =2 : Z ˜S 

Z S =1, Z D = Z S . (7.61) 

For the following, it is crucial that the theory is renormalisable, i.e., a finite 

number of reparametrisations suffice to formally rid it of all UV divergences. 

Indeed, for the relaxational models A and B, and the static Ginzburg–Landau– 

Wilson Hamiltonian (7.7), all higher vertex function beyond the four-point 

function are UV-convergent near d c , and there are only the three independent 

static renormalisation factors Z S , Z τ ,andZ u , and in addition Z D for nonconserved 

order parameter dynamics. As we shall see, these directly translate 

into the two independent static critical exponents and the unrelated dynamic 

scaling exponent z for model A; for model B with conserved order parameter, 

Eq. (7.61) will yield a scaling relation between z and η. 

In order to explicitly determine the renormalisation constants, we need 

to ensure that we stay away from the IR-singular regime. This is guaranteed 

by selecting as normalisation point either τ R = 1 (i.e., Z τ τ = µ 2 )orq = µ. 

Inevitably therefore, the renormalised theory depends on the corresponding 

arbitrary momentum scale µ. Since there are no fluctuation contributions to 

order u to either ∂Γ (1,1) (q, 0)/∂q 2 or ∂Γ (1,1) (0,ω)/∂ω (at τ R = 1), we find 

Z S =1andZ D = 1 within the one-loop approximation. Expressions (7.55) 

and (7.50) then yield with the formula (7.53) 

Z τ =1− n +2 

6 

u R 

ɛ 

, Z u =1− n +8 

6 

u R 

ɛ 

. (7.62) 

To two-loop order, we may infer the field renormalisation Z S from the static 

susceptibility as the singular contributions to ∂χ R (q, 0)/∂q 2 | q=0 ,andZ D for 

model A, through a somewhat lengthy calculation [13], from either Γ (2,0) 

R 

(0, 0) 

or Γ (1,1) 

R 

(0,ω), with the results 

Z S =1+ n +2 

144 

u 2 R 

ɛ 

, a =0 : Z D =1− n +2 

144 

7.1.5 Scaling Laws and Critical Exponents 

( 

6ln 4 3 − 1 ) u 

2 

R 

ɛ 

. (7.63) 

We now wish to related the renormalised vertex functions at different inverse 

length scales µ. This is accomplished by simply recalling that the unrenormalised 

vertex functions obviously do not depend on µ, 

0=µ d 

dµ Γ (Ñ,N) (D, τ, u) =µ d [ 

dµ 

ZÑ/2 ˜S 

] 

Z N/2 

S 

Γ (Ñ,N) 

R 

(µ, D R ,τ R ,u R ) . (7.64) 

In the second step, the bare quantities have been replaced with their renormalised 

counterparts. The innocuous statement (7.64) then implies a very 

nontrivial partial differential equation for the renormalised vertex functions, 

the desired renormalisation group equation,


[ 

µ ∂ 

∂µ + Ñγ˜S + Nγ S 

∂ 

+ γ D D R + γ τ τ R 

2 

∂D R 

∂ 

∂τ R 

+ β u 

] 

∂ 

∂u R 

× Γ (Ñ,N) 

R 

(µ, D R ,τ R ,u R )=0. (7.65) 

Here we have defined Wilson’s flow functions (the index “0” indicates that 

the derivatives with respect to µ are to be taken with fixed unrenormalised 

parameters) 

γ ˜S 

= µ ∂ 

∂µ ∣ ln Z ˜S 

, γ S = µ ∂ 

0 

∂µ ∣ ln Z S , (7.66) 

0 

γ τ = µ ∂ 

∂µ ∣ ln(τ R /τ) =−2+µ ∂ 

0 

∂µ ∣ ln Z τ , (7.67) 

0 

γ D = µ ∂ 

∂µ ∣ ln(D R /D) = 1 ( ) 

γS − γ 

0 

2 

˜S , (7.68) 

where we have used the relation (7.59); for model B, Eq. (7.61) gives in addition 

γ D = γ S = −γ ˜S 

. (7.69) 

We have also introduced the RG beta function for the nonlinear coupling u, 

β u = µ 

∂ 

( 

∂µ ∣ u R = u R d − 4+µ 

∂ 

) 

0 

∂µ ∣ ln Z u . (7.70) 

0 

Explicitly, Eqs. (7.63) and (7.62) yield to lowest nontrivial order, with ɛ = 

4 − d, 

γ S = − n +2 

72 u2 R + O(u 3 R) , (7.71) 

a =0 : γ D = n +2 (6ln 4 ) 

72 3 − 1 u 2 R + O(u 3 R) , (7.72) 

γ τ = −2+ n +2 u R + O(u 2 

6 

R) , (7.73) 

[ 

β u = u R −ɛ + n +8 

] 

u R + O(u 2 

6 

R) . (7.74) 

In the RG equation for the renormalised dynamic susceptibility, Eq. (7.60) 

tells us that the second term in Eq. (7.65) is to be replaced with −γ S . Its explicit 

dependence on the scale µ can be factored out via χ R (µ, D R ,τ R ,u R , q,ω) 

= µ −2 ˆχ R 

( 

τR ,u R , q/µ, ω/D R µ 2+a) , see Eq. (7.55), whence 

[ 

−2 − γ S + γ D D R 

∂ 

∂D R 

+ γ τ τ R 

∂ 

∂τ R 

+ β u 

] 

∂ 

ˆχ R (D R ,τ R ,u R )=0. (7.75) 

∂u R 

This linear partial differential equation is readily solved by means of the 

method of characteristics, as is Eq. (7.65) for the vertex functions. The idea


is to find a curve parametrisation µ(l) =µl in the space spanned by the 

parameters ˜D, ˜τ, andũ such that 

l d ˜D(l) 

dl 

= ˜D(l) γ D (l) , l d˜τ(l) 

dl 

= ˜τ(l) γ τ (l) , l dũ(l) 

dl 

= β u (l) , (7.76) 

with initial values D R , τ R ,andu R , respectively at l =1.Thefirst-order ordinary 

differential equations (7.76), with γ D (l) =γ D (ũ(l)) etc. define running 

couplings that describe how the parameters of the theory change under scale 

transformations µ → µl. The formal solutions for ˜D(l) and˜τ(l) read 

˜D(l) =D R exp 

[ ∫ l 

1 

γ D (l ′ ) dl′ 

l ′ ] 

, ˜τ(l) =τ R exp 

[ ∫ l 

1 

γ τ (l ′ ) dl′ 

l ′ ] 

. (7.77) 

For the function ˆχ(l) =ˆχ R ( ˜D(l), ˜τ(l), ũ(l)), we then obtain another ordinary 

differential equation, namely 

which is solved by 

l dˆχ(l) 

dl 

ˆχ(l) =ˆχ(1) l 2 exp 

=[2+γ S (l)] ˆχ(l) , (7.78) 

[ ∫ l 

1 

γ S (l ′ ) dl′ 

l ′ ] 

. (7.79) 

Collecting everything, we finally arrive at 

[ 

χ R (µ, D R ,τ R ,u R , q,ω)=(µl) −2 exp 

× ˆχ R 

( 

− 

∫ l 

1 

γ S (l ′ ) dl′ 

l ′ ] 

˜τ(l), ũ(l), |q| 

µl , ω 

˜D(l)(µl) 2+a ) 

. (7.80) 

The solution (7.80) of the RG equation (7.75), along with the flow equations 

(7.76), (7.77) for the running couplings tell us how the dynamic susceptibility 

depends on the (momentum) scale µlat which we consider the theory. 

Similar relations can be obtained for arbitrary vertex functions by solving the 

associated RG equations (7.65) [13]. The point here is that the right-hand side 

of Eq. (7.80) may be evaluated outside the IR-singular regime, by fixing one 

of its arguments at a finite value, say |q|/µ l = 1. The function ˆχ R is regular, 

and can be calculated by means of perturbation theory. A scale-invariant 

regime is characterised by the renormalised nonlinear coupling u R becoming 

independent of the scale µl,orũ(l) → u ∗ = const. For an RG fixed point to 

be infrared-stable, we thus require 

β u (u ∗ )=0, β ′ u(u ∗ ) > 0 , (7.81)


since Eq. (7.76) then implies that ũ(l → 0) → u ∗ . Taking the limit l → 0 

thus provides the desired mapping of physical observables such as (7.80) onto 

the critical region. In the vicinity of an IR-stable RG fixed point, Eq. (7.77) 

yields the power laws ˜D(l) ≈ D R l γ∗ D , where γ 

∗ 

D = γ D (l → 0) = γ D (u ∗ ), etc. 

Consequently, Eq. (7.80) reduces to 

( 

χ R (τ R , q,ω) ≈ µ −2 l −2−γ∗ S ˆχR τ R l γ∗ τ ,u ∗ , |q| 

) 

µl , ω 

, (7.82) 

D R µ 2+a l 2+a+γ∗ D 

and upon matching l = |q|/µ we recover the dynamic scaling law (7.12) with 

the critical exponents 

η = −γ ∗ S , ν = −1/γ ∗ τ , z =2+a + γ ∗ D . (7.83) 

To one-loop order, we obtain from the RG beta function (7.74) 

u ∗ H = 

6 ɛ 

n +8 + O(ɛ2 ) . (7.84) 

Here we have indicated that our perturbative expansion for small u has effectively 

turned into a dimensional expansion in ɛ = d c − d. In dimensions 

d0. With 

Eqs. (7.71) and (7.73), the identifications (7.83) then give us explicit results 

for the static scaling exponents, as mere functions of dimension d =4− ɛ and 

the number of order parameter components n, 

η = n +2 

2(n +8) 2 ɛ2 + O(ɛ 3 ) , 

1 

ν 

=2− 

n +2 

n +8 ɛ + O(ɛ2 ) . (7.85) 

For model A with nonconserved order parameter, the two-loop result (7.72) 

yields the independent dynamic critical exponent 

a =0 : z =2+cη , c=6ln 4 − 1+O(ɛ) ; (7.86) 

3 

for model B with conserved order parameter, instead γD ∗ = γ∗ S = −η, whence 

we arrive at the exact scaling relation 

a =2 : z =4− η. (7.87) 

In dimensions d > d c = 4, the Gaussian fixed point u ∗ 0 = 0 is stable 

(β ′ u(0) = −ɛ >0). Therefore all anomalous dimensions disappear, i.e., γ ∗ S = 

0=γ ∗ D and γ∗ τ = −2, and we are left with the mean-field critical exponents 

η 0 =0,ν 0 =1/2, and z 0 =2+a. Precisely at the upper critical dimension 

d c = 4, the RG flow equation for the nonlinear coupling becomes 

which is solved by 

l dũ(l) 

dl 

= n +8 

6 

( 

ũ(l) 2 + O ũ(l) 3) , (7.88)


ũ(l) = 

u R 

1 − n+8 

6 

u R ln l . (7.89) 

In four dimensions, ũ(l) → 0, but only logarithmically slowly, which causes 

logarithmic corrections to the mean-field critical power laws. For example, 

upon inserting Eq. (7.89) into the flow equation (7.76), one finds ˜τ(l) ∼ 

τ R l −2 (ln |l|) −(n+2)/(n+8) ; with ˜τ(l = ξ −1 )=O(1), iterative inversion yields 

ξ(τ R ) ∼ τ −1/2 

R 

(ln τ R ) (n+2)/2(n+8) . (7.90) 

This concludes our derivation of asymptotic scaling laws for the critical 

dynamics of the purely relaxational models A and B, and the explicit computation 

of the scaling exponents in powers of ɛ = d c − d. In the following 

sections, I will briefly sketch how the response functional formalism and the 

dynamic renormalisation group can be employed to study the critical dynamics 

of systems with reversible mode-coupling terms, the “ageing’ behaviour 

induced by quenching from random initial conditions to the critical point, the 

effects of violating the detailed balance constraints on universal dynamic critical 

properties, and the generically scale-invariant features of nonequilibrium 

systems such as driven diffusive Ising lattice gases. 

7.1.6 Critical Dynamics with Reversible Mode-Couplings 

In the previous chapters, we have assumed purely relaxational dynamics for 

the order parameter, see Eq. (7.16). In general, however, there are also reversible 

contributions to the systematic force terms F α that enter its Langevin 

equation [7,24]. Consider the Hamiltonian dynamics of microscopic variables, 

say, local spin densities, at T =0:∂ t Sm(x, α t) = {H[S m ],Sm(x, α t)}. Here, 

the Poisson brackets {A, B} constitute the classical analog of the quantummechanical 

commutator 

i 

[A, B] (correspondence principle). Upon coarsegraining, 

the microscopic variables Sm α become the mesoscopic hydrodynamic 

fields S α . Since the set of slow modes should provide a complete description 

of the critical dynamics, we may formally expand 

{ 

} 

H[S],S α (x) = 

∫ 

d d x ∑ ′ β 

δH[S] 

δS β (x ′ ) Qβα (x ′ , x) , (7.91) 

with the mutual Poisson brackets of the hydrodynamic variables 

{ 

} 

Q αβ (x, x ′ )= S α (x),S β (x ′ ) = −Q βα (x ′ , x) . (7.92) 

By inspection of the associated Fokker–Planck equation, one may then 

establish an additional equilibrium condition in order for the time-dependent 

probability distribution to reach the canonical limit (7.6): P[S, t] →P eq [S] as 

t →∞provided the probability current is divergence-free in the space spanned 

by the stochastic fields S α (x):


∫ 

d d x ∑ δ 

( 

δS α F α 

(x) 

rev[S] e −H[S]/kBT =0. 

α 

(7.93) 

It turns out that this equilibrium condition is often more crucial than the 

Einstein relation (7.17). In order to satisfy Eq. (7.93) at T ≠ 0, we must 

supplement Eq. (7.91) by a finite-temperature correction, whereupon the reversible 

mode-coupling contributions to the systematic forces become 

∫ 

Frev[S](x) α =− d d x ∑ ′ β 

[ 

Q αβ (x, x ′ ) δH[S] 

δS β (x ′ ) − k BT δQαβ (x, x ′ ] 

) 

δS β (x ′ , 

) 

(7.94) 

and the complete coupled set of stochastic differential equations reads 

∂S α (x,t) 

∂t 

= F α rev[S](x,t) − D α (i∇) aα δH[S] 

δS α (x,t) + ζα (x,t) , (7.95) 

where as before the D α denote the relaxation coefficients, and a α =0or2 

respectively for nonconserved and conserved modes. 

As an instructive example, let us consider the Heisenberg model for 

isotropic ferromagnets, H[{S j }]=− 1 ∑ N 

2 j,k=1 J jk S j · S k , where the spin operators 

satisfy the usual commutation relations [Sj α,Sβ k ]=i ∑ γ ɛαβγ S γ j δ jk. 

The corresponding Poisson brackets for the magnetisation density read 

Q αβ (x, x ′ )=−g ∑ γ 

ɛ αβγ S γ (x) δ(x − x ′ ) , (7.96) 

where the purely dynamical coupling g incorporates various factors that 

emerge upon coarse-graining and taking the continuum limit. The second 

contribution in Eq. (7.94) vanishes, since it reduces to a contraction of the 

antisymmetric tensor ɛ αβγ with the Kronecker symbol δ βγ , whence we arrive 

at the Langevin equations governing the critical dynamics of the three order 

parameter components for isotropic ferromagnets [25] 

∂S(x,t) 

∂t 

= −g S(x,t) × δH[S] 

δS(x,t) + D δH[S] 

∇2 + ζ(x,t) , (7.97) 

δS(x,t) 

with 〈ζ(x,t)〉 = 0. Since [H[{S j }], ∑ k Sα k 

] = 0, the total magnetisation is 

conserved, whence the noise correlators should be taken as 

〈 

ζ α (x,t) ζ β (x ′ ,t ′ ) 〉 = −2Dk B T ∇ 2 δ(x − x ′ ) δ(t − t ′ ) δ αβ . (7.98) 

The vector product term in Eq. (7.97) describes the spin precession in the local 

effective magnetic field δH[S]/δS, which includes a contribution induced by 

the exchange interaction. 

The Langevin equation (7.97) and (7.98) with the Hamiltonian (7.7) for 

n = 3 define the so-called model J [7]. In addition to the model B response


functional (7.35) and (7.36) with a = 2 (setting k B T = 1 again), the reversible 

force in Eq. (7.97) leads to an additional contribution to the action 

∫ ∫ 

A mc [ ˜S,S]=−g d d x dt ∑ ɛ αβγ ˜Sα S β ( ∇ 2 S γ + h γ) , (7.99) 

α,β,γ 

which gives rise to an additional mode-coupling vertex, as depicted in Fig. 7.5(a). 

Power counting yields the scaling dimension [g] =µ 3−d/2 for the associated 

coupling strength, whence we expect a dynamical upper critical dimension 

d ′ c = 6. However, since we are investigating a system in thermal equilibrium, 

we can treat its thermodynamics and static properties separately from its dynamics. 

Obviously therefore, the static critical exponents must still be given 

(to lowest nontrivial order and for d


where B d = Γ (4 − d/2)/2 d dπ d/2 , Eq. (7.101) implies the identity [9] 

Z g = Z S . (7.103) 

For the RG beta function associated with the effective coupling entering the 

loop corrections, we thus infer 

β f = µ ∂ 

∂µ ∣ f R = f R (d − 6+γ S − 2 γ D ) . (7.104) 

0 

Consequently, at any nontrivial IR-stable RG fixed point 0


7.1.7 Critical Relaxation, Initial Slip, and Ageing 

We begin with a brief discussion of the coarsening dynamics of systems described 

by model A/B kinetics that are rapidly quenched from a disordered 

state at T ≫ T c to the critical point T ≈ T c [27, 28]. The situation may be 

modeled as a relaxation from Gaussian random initial conditions, i.e., the 

probability distribution for the order parameter at t = 0 can be taken as 

( 

P[S, t =0]∝ e −H0[S] =exp − ∆ ∫ 

d d x ∑ ) 

[S α (x, 0) − a α (x)] 2 , (7.108) 

2 

α 

where the functions a α (x) specify the most likely initial configurations. Power 

counting for the parameter ∆ gives [∆] =µ 2 , whence it is a relevant perturbation 

that will flow to ∆ →∞under the RG. Asymptotically, therefore, the 

system will be governed by sharp Dirichlet boundary conditions. Whereas the 

response propagators remains a causal function of the time difference between 

applied perturbation and effect, G 0 (q,t−t ′ )=Θ(t−t ′ )e −Dqa (r+q 2 )(t−t ′) ,see 

Eq. (7.23), time translation invariance is broken by the initial state in the 

Dirichlet correlator of the Gaussian model, 

C D (q; t, t ′ )= 1 ( 

r + q 2 e −Dqa (r+q 2 ) |t−t ′| ) − e −Dqa (r+q 2 )(t+t ′ ) 

. (7.109) 

Away from criticality, i.e., for r>0andq ≠ 0, temporal correlations decay 

exponentially fast, and the system quickly approaches the stationary equilibrium 

state. However, as T → T c , the equilibration time diverges according to 

t c ∼|τ| −zν →∞, and the system never reaches thermal equilibrium. Twotime 

correlation functions will then depend on both times separately, in a 

specific manner to be addressed below, a phenomenon termed critical “ageing” 

(for more details, see Refs. [29, 30]). 

The field-theoretic treatment of the model A/B dynamical action (7.35), 

(7.36) with the initial term (7.108) follows the theory of boundary critical 

phenomena [31]. However, it turns out that additional singularities on the 

temporal “surface” at t + t ′ = 0 appear only for model A, and can be incorporated 

into a single new renormalisation factor; to one-loop order, one 

finds [27, 28] 

a =0 : ˜Sα R (x, 0) = (Z 0 Z ˜S) 1/2 ˜Sα (x, 0) , Z 0 =1− n +2 

6 

u R 

ɛ 

. (7.110) 

This in turn leads to a single independent critical exponent associated with 

the initial time relaxation, the initial slip exponent, which becomes for the 

purely relaxational models A and B with nonconserved and conserved order 

parameter: 

a =0 : θ = γ∗ 0 

2 z = n +2 

4(n +8) ɛ + O(ɛ2 ) , a =2 : θ =0. (7.111)


In order to obtain the short-time scaling laws for the dynamic response and 

correlation functions in the ageing limit t ′ /t → 0, one requires additional 

information that can be garnered from the short-distance operator product 

expansion for the fields, 

t → 0: ˜S(x,t)=˜σ(t) ˜S0 (x) , S(x,t)=σ(t) ˜S 0 (x) . (7.112) 

Subsequent analysis then yields eventually [27, 28] 

χ(q; t, t ′ → 0) = |q| z−2+η ( t 

t ′ ) θ 

ˆχ 0 (q ξ,|q| z Dt) , (7.113) 

C(q; t, t ′ → 0) = |q| −2+η ( t 

t ′ ) θ−1 

Ĉ0 (q ξ,|q| z Dt) , (7.114) 

and for the time dependence of the mean order parameter 

( ) 

〈S(t)〉 = S 0 t θ′ Ŝ S 0 t θ′ +β/zν 

, (7.115) 

a =0 : θ ′ = θ − z − 2+η 

z 

, a =2 : θ ′ = θ =0. (7.116) 

One may also compute the universal fluctuation–dissipation ratios in this 

nonequilibrium ageing regime [29, 30]. It emerges, though, that these depend 

on the quantity under investigation, which prohibits a unique definition of an 

effective nonequilibrium temperature for critical ageing. The method sketched 

above can be extended to models with reversible mode-couplings [32]. For 

model J capturing the critical dynamics of isotropic ferromagnets, one finds 

θ = z − 4+η 

z 

= − 6 − d − η 

d +2− η ; (7.117) 

in systems where a nonconserved order parameter is dynamically coupled to 

other conserved modes, the initial slip exponent θ is actually not a universal 

number, but depends on the width of the initial distribution [32]. 

7.1.8 Nonequilibrium Relaxational Critical Dynamics 

Next we address the question [33], What happens if the detailed balance conditions 

(7.17) and (7.93) are violated? To start, we change the noise strength 

D → ˜D in the purely relaxational models A and B, which (in our units) violates 

the Einstein relation (7.17). However, this modification can obviously be 

absorbed into a rescaled effective temperature, k B T → k B T ′ = ˜D/D. Formally 

this is established by means of the dynamical action (7.34), which now reads 

∫ ∫ 

A[ ˜S,S] = d d x dt ∑ ˜S 

[∂ α t S α + D (i∇) a ( r − ∇ 2) S α 

α 

− ˜D (i∇) a ˜Sα + D u 6 (i∇)a S ∑ ] 

α S β S β . (7.118) 

β


√ 

Upon simple rescaling ˜S α → ˜S ′α = ˜S α ˜D/D, S α → S ′α = S 

√D/ α ˜D, the 

response functional (7.118) recovers its equilibrium form, albeit with modified 

nonlinear coupling u → ũ = u ˜D/D. However, the universal asymptotic 

properties of these models are governed by the Heisenberg fixed point (7.84), 

and the specific value of the (renormalised) coupling, which only serves as 

the initial condition for the RG flow, does not matter. In fact, the relaxational 

dynamics of the kinetic Ising model with Glauber dynamics (model 

A with n = 1) is known to be quite stable against nonequilibrium perturbations 

[34, 35], even if these break the Ising Z 2 symmetry [36]. For model J 

the above rescaling modifies in a similar manner √ merely the mode-coupling 

strength in Eq. (7.99), namely g → ˜g = g ˜D/D [37]. Again, since the dynamic 

critical behaviour is governed by the universal fixed point (7.107), thermal 

equilibrium becomes effectively restored at criticality. More generally, it 

has been established that isotropic detailed balance violations do not affect 

the universal properties in other models for critical dynamics that contain 

additional conserved variables either: the equilibrium RG fixed points tend to 

be asymptotically stable [33]. 

In systems with conserved order parameter, however, we may in addition 

introduce spatially anisotropic violations of Einstein’s relation; for example, in 

model B one can allow for anisotropic relaxation −D ∇ 2 →−D ⊥ ∇ 2 ⊥ −D ‖ ∇ 2 ‖ , 

with different rates in two spatial subsectors and concomitantly anisotropic 

noise correlations − ˜D ∇ 2 →−˜D ⊥ ∇ 2 ⊥ − ˜D ‖ ∇ 2 ‖. We have thus produced a 

truly nonequilibrium situation provided ˜D ⊥ /D ⊥ ≠ ˜D ‖ /D ‖ , which we may 

interpret as having effectively coupled the longitudinal and transverse spatial 

sectors to heat baths with different temperatures T ⊥


Quite remarkably, the Langevin equation (7.119) can be derived as an equilibrium 

diffusive relaxational kinetics 

∂S α (x,t) 

∂t 

= D ∇ 2 ⊥ 

from an effective long-range Hamiltonian 

∫ 

H eff [S] = 

+ ũ 

4! 

δH eff [S] 

δS α (x,t) + ζα (x,t) (7.121) 

d d q q 2 ⊥ (r + q2 ⊥ )+c q2 ∑ 

‖ 

∣ S α 

(2π) d 2 q 2 (q) ∣ 2 

⊥ 

α 

∫ 

d d x ∑ [S α (x)] 2 [S β (x)] 2 . (7.122) 

α,β 

Power counting gives [ũ] =µ 4−d ‖−d : the spatial anisotropy suppresses longitudinal 

fluctuations and lower the upper critical dimension to d c =4−d ‖ .The 

anisotropic correlations encoded in Eq. (7.122) also reduce the lower critical 

dimension and affect the nature of the ordered phase [39, 41]. 

The scaling law for, e.g., the dynamic response function takes the form 

( √ ) 

τ c 

χ(τ ⊥ , q ⊥ , q ‖ ,ω)=|q ⊥ | −2+η ˆχ 

|q ⊥ | , |q‖ | 

1/ν |q ⊥ | 1+∆ , ω 

D |q ⊥ | z , (7.123) 

where we have introduced a new anisotropy exponent ∆. Since the nonlinear 

coupling ũ only affects the transverse sector, we find to all orders in the 

perturbation expansion: 

Γ (1,1) (q ⊥ =0, q ‖ ,ω)=iω + Dcq 2 ‖ , (7.124) 

and consequently obtain the Z factor identity 

Z c = Z −1 

D 

= Z−1 S 

, (7.125) 

which at any IR-stable RG fixed point implies the exact scaling relations 

z =4− η, 

∆=1− γ∗ c 

2 =1− η 2 = z 2 − 1 , (7.126) 

whereas the scaling exponents for the longitudinal sector read 

z ‖ = 

z 

1+∆ =2, ν ‖ = ν (1 + ∆) = ν (4 − η) . (7.127) 

2 

As for the equilibrium model B, the only independent critical exponents to 

be determined are η and ν. To one-loop order, only the combinatorics of the 

Feynman diagrams (see Fig. 7.2) enters their explicit values, whence one finds 

for d


η =0+O(ɛ 2 ) , 

1 

ν 

=2− 

n +2 

n +8 ɛ + O(ɛ2 ) , (7.128) 

albeit with different ɛ =4−d−d ‖ . To two-loop order, however, the anisotropy 

manifestly affects the evaluation of the loop contributions, and the value for 

η deviates from the expression in Eq. (7.85) [40]. 

Interestingly, an analogously constructed nonequilibrium two-temperature 

model J with reversible mode-coupling vertex cannot be cast into a form that 

is equivalent to an equilibrium system, for owing to the emerging anisotropy, 

the condition (7.93) cannot be satisfied. A one-loop RG analysis yields a runaway 

flow, and no stable RG fixed point is found [38]. Similar behaviour ensues 

in other anisotropic nonequilibrium variants of critical dynamics models with 

conserved order parameter; the precise interpretation of the apparent instability 

is as yet unclear [33]. 

7.1.9 Driven Diffusive Systems 

Finally, we wish to consider Langevin representations of genuinely nonequilibrium 

systems, namely driven diffusive lattice gases (for a comprehensive 

overview, see Ref. [42]). First we address the coarse-grained continuum version 

of the asymmetric exclusion process, i.e., hard-core repulsive particles that hop 

preferentially in one direction. We describe this system in terms of a conserved 

particle density, whose fluctuations we denote with S(x,t), such that 〈S〉 =0, 

obeying a continuity equation ∂ t S(x,t)+∇ · J(x,t) = 0. We assume the system 

to be driven along the “‖’ direction; in the transverse sector (of dimension 

d ⊥ = d − 1) we thus just have a noisy diffusion current J ⊥ = −D ∇ ⊥ S + η, 

whereas there is a nonlinear term, stemming from the hard-core interactions, 

in the current along the direction of the external drive, with J 0 ‖ =const.: J ‖ = 

J 0 ‖ −Dc∇ ‖ S − 1 2 Dg S2 +η ‖ . For the stochastic currents, we assume Gaussian 

white noise 〈η i 〉 =0=〈η ‖ 〉 and 〈η i (x,t) η j (x ′ ,t ′ )〉 =2Dδ(x − x ′ ) δ(t − t ′ ) δ ij , 

〈 

η‖ (x,t) η ‖ (x ′ ,t ′ ) 〉 =2D ˜cδ(x − x ′ ) δ(t − t ′ ). Notice that since we are not in 

thermal equilibrium, Einstein’s relation need not be fulfilled. We can however 

always rescale the field to satisfy it in the transverse sector; the ratio w =˜c/c 

then measures the deviation from equilibrium. These considerations yield the 

generic Langevin equation for the density fluctuations in driven diffusive systems 

(DDS) [43, 44] 

∂S(x,t) 

∂t 

( ) 

= D ∇ 2 ⊥ + c ∇ 2 ‖ S(x,t)+ Dg 

2 ∇ ‖S(x,t) 2 + ζ(x,t) , (7.129) 

with conserved noise ζ = −∇ ⊥ · η −∇ ‖ η ‖ , where 〈ζ〉 =0and 

( ) 

〈ζ(x,t) ζ(x ′ ,t ′ )〉 = −2D ∇ 2 ⊥ +˜c ∇ 2 ‖ δ(x − x ′ ) δ(t − t ′ ) . (7.130) 

Notice that the drive term ∝ g breaks both the system’s spatial reflection 

symmetry and the Ising Z 2 symmetry S →−S.


The corresponding Janssen–De Dominicis response functional (7.34) reads 

∫ ∫ [ 

A[ ˜S,S] = d d x dt ˜S ∂S 

( ) 

∂t − D ∇ 2 ⊥ + c ∇ 2 ‖ S 

] 

( ) 

+ D ∇ 2 ⊥ +˜c ∇ 2 Dg 

‖ ˜S − 

2 ∇ ‖ S 2 . (7.131) 

It describes a “massless” theory, hence we expect the system to be generically 

scale-invariant, without the need to tune it to a special point in parameter 

space. The nonlinear drive term will induce anomalous scaling in the drive 

direction, different from ordinary diffusive behaviour. In the transverse sector, 

however, we have to all orders in the perturbation expansion simply 

Γ (1,1) (q ⊥ ,q ‖ =0,ω)=iω +D q 2 ⊥ , Γ (2,0) (q ⊥ ,q ‖ =0,ω)=−2D q 2 ⊥ , (7.132) 

since the nonlinear three-point vertex, which is of the form depicted in 

Fig. 7.5(a), is proportional to iq ‖ . Consequently, 

which immediately implies 

Z ˜S 

= Z S = Z D =1, (7.133) 

η =0, z =2. (7.134) 

Moreover, the nonlinear coupling g itself does not renormalise either as a 

consequence of Galilean invariance. Namely, the Langevin equation (7.129) 

and the action (7.131) are left invariant under Galilean transformations 

S ′ (x ′ ⊥,x ′ ‖ ,t′ )=S(x ⊥ ,x ‖ − Dgv t, t) − v ; (7.135) 

thus, the boost velocity v must scale as the field S under renormalisation, and 

since the product Dgv must be invariant under the RG, this leaves us with 

Z g = Z −1 

D 

Z−1 S 

=1. (7.136) 

The effective nonlinear coupling governing the perturbation expansion in 

terms of loop diagrams turns out to be g 2 /c 3/2 ; if we define its renormalised 

counterpart as 

v R = Zc 3/2 vC d µ d−2 , (7.137) 

with the convenient choice C d = Γ (2 − d/2)/2 d−1 π d/2 , we see that the associated 

RG beta function becomes 

β v = v R 

(d − 2 − 3 ) 

2 γ c . (7.138) 

At any nontrivial RG fixed point 0 < v ∗ < ∞, therefore γ ∗ c = 2 3 

(d − 2). 

We thus infer that below the upper critical dimension d c = 2 for DDS, the 

longitudinal scaling exponents are fixed by the system’s symmetry [43, 44],


∆ = − γ∗ c 

2 = 2 − d 

3 

, z ‖ = 2 

1+∆ = 6 

5 − d . (7.139) 

An explicit one-loop calculation for the two-point vertex functions, see 

Fig. 7.5(b) and (c), yields 

γ c = − v R 

16 (3 + w R) , γ˜c = − v R 

( 

3w 

−1 

R 

32 

R) 

, (7.140) 

β w = w R (γ˜c − γ c )=− v R 

32 (w R − 1) (w R − 3) . (7.141) 

This establishes that in fact the fixed point w ∗ = 1 is IR-stable (provided 0 < 

v ∗ < ∞), which means that asymptotically the Einstein relation is satisfied 

in the longitudinal sector as well [43]. 

In this context, it is instructive to make an intriguing connection with the 

noisy Burgers equation [45], describing simplified fluid dynamics in terms of 

a velocity field u(x,t): 

∂u(x,t) 

+ Dg 

∂t 2 ∇ [ u(x,t) 2] = D ∇ 2 u(x,t)+ζ(x,t) , (7.142) 

〈ζ i 〉 =0, 〈ζ i (x,t) ζ j (x ′ ,t ′ )〉 = −2D ∇ i ∇ j δ(x − x ′ ) δ(t − t ′ ) . (7.143) 

For Dg = 1, the nonlinearity is just the usual fluid advection term. In one 

dimension, the Burgers equation (7.142) becomes identical with the DDS 

Langevin equation (7.129), so we immediately infer its anomalous dynamic 

critical exponent z ‖ =3/2. At least in one dimension therefore, it should represent 

an equilibrium system which asymptotically approaches the canonical 

distribution (7.6), where the Hamiltonian is simply the fluid’s kinetic energy 

(andwehavesetk B T = 1). So let us check the equilibrium condition (7.93) 

with P eq [u] ∝ exp [ ∫ 

− 1 2 u(x) 2 d d x ] : 

∫ 

d d δ 

x 

δu(x,t) · [∇u(x,t) 2] e − 1 ∫ 

2 u(x ′ ,t) 2 d d x ′ 

= 

∫ [2∇ 

· u(x,t) − u(x,t) · ∇u(x,t) 

2 ] d d x e − 1 2 

∫ 

u(x ′ ,t) 2 d d x ′ . 

With appropriate boundary conditions, the first term here vanishes, but the 

second one does so only in d = 1: − ∫ u (du 2 /dx)dx = ∫ ∫ u 2 (du/dx)dx = 

1 

3 (du 3 /dx)dx = 0. Driven diffusive systems in one dimension are therefore 

subject to a “hidden” fluctuation–dissipation theorem. 

To conclude this part on Langevin dynamics, let us briefly consider the 

driven model B or critical DDS [42], which corresponds to a driven Ising lattice 

gas near its critical point. Here, a conserved scalar field S undergoes a secondorder 

phase transition, but similar to the randomly driven case, again only the 

transverse sector is critical. Upon adding the DDS drive term from Eq. (7.129) 

to the Langevin equation (7.119), we obtain

∂S(x,t) 

∂t 


[ 

= D ∇ 2 ( ) ] 

⊥ r − ∇ 

2 

⊥ + c ∇ 

2 

‖ S(x,t)+ D ũ 

6 ∇2 ⊥ S(x,t) 3 

+ Dg 

2 ∇ ‖ S(x,t) 2 + ζ(x,t) , (7.144) 

with the (scalar) noise specified in Eq. (7.120). The response functional thus 

becomes 

∫ ∫ [ 

A[ ˜S,S] = d d x dt ˜S ∂S 

[ 

∂t − D ∇ 2 ( ) ] 

⊥ r − ∇ 

2 

⊥ + c ∇ 

2 

‖ S 

( 

+D ∇ 2 ˜S ⊥ − ũ 

6 ∇2 ⊥ S 3 − g ) ] 

2 ∇ ‖ S 2 . (7.145) 

Power counting gives [g 2 ] = µ 5−d , so the upper critical dimension here is 

d c = 5, and [ũ] =µ 3−d . The nonlinearity ∝ ũ is thus irrelevant and can 

be omitted if we wish to determine the asymptotic universal scaling laws; 

but recall that it is responsible for the phase transition in the system. The 

remaining vertex is then proportional to iq ‖ , whence Eqs. (7.132) and (7.133) 

hold for critical DDS as well, and the transverse critical exponents are just 

those of the Gaussian model B, 

η =0, ν =1/2 , z =4. (7.146) 

In addition, Galilean invariance with respect to Eq. (7.135) and therefore 

Eq. (7.136) hold as before. With the renormalised nonlinear drive strength 

defined similarly to Eq. (7.137), but a different geometric constant and the 

scale factor µ d−5 , the associated RG beta function reads 

β v = v R 

(d − 5 − 3 ) 

2 γ c , (7.147) 

which again allows us to determine the longitudinal scaling exponents to all 

orders in perturbation theory, for d


7.2 Reaction–Diffusion Systems 

We now turn our attention to stochastic interacting particle systems, whose 

microscopic dynamics is defined through a (classical) master equation. Below, 

we shall see how the latter can be mapped onto a stochastic quasi-Hamiltonian 

in a second-quantised bosonic operator representation [10–12]. Taking the 

continuum limit on the basis of coherent-state path integrals then yields a field 

theory action that may be analysed by the very same RG methods as described 

before in Subsects. 7.1.3–7.1.5 (for more details, see the recent overview [12]). 

7.2.1 Chemical Reactions and Population Dynamics 

Our goal is to study systems of “particles” A,B,... that propagate through 

hopping to nearest neighbors on a d-dimensional lattice, or via diffusion in 

the continuum. Upon encounter, or spontaneously, with given stochastic rates, 

these particles may undergo species changes, annihilate, or produce offspring. 

At large densities, the characteristic time scales of the kinetics will be governed 

by the reaction rates, and the system is said to be reaction-limited. In contrast, 

at low densities, any reactions that require at least two particles to be in 

proximity will be diffusion-limited: the basic time scale will be set by the 

hopping rate or diffusion coefficient. 

As a first approximation to the dynamics of such “chemical” reactions, 

let us assume homogeneous mixing of each species. We may then hope to be 

able to capture the kinetics in terms of rate equations for each particle concentration 

or mean density. Note that such a description neglects any spatial 

fluctuations and correlations in the system, and is therefore in character a 

mean-field approximation. As a first illustration consider the annihilation of 

k−l >0 particles of species A in the irreversible kth-order reaction kA→ lA, 

with rate λ. The corresponding rate equation employs a factorisation of the 

probability of encountering k particles at the same point to simply the kth 

power of the concentration a(t), 

ȧ(t) =−(k − l) λa(t) k . (7.149) 

This ordinary differential equation is readily solved, with the result 

k =1 : a(t) =a(0) e −λt , (7.150) 

k ≥ 2: a(t) = [ a(0) 1−k +(k − l)(k − 1) λt ] −1/(k−1) 

. (7.151) 

For simple “radioactive” decay (k = 1), we of course obtain an exponential 

time dependence, as appropriate for statistically independent events. For 

pair (k = 2) and higher-order (k ≥ 3) processes, however, we find algebraic 

long-time behaviour, a(t) → (λt) −1/(k−1) , with an amplitude that becomes 

independent of the initial density a(0). The absence of a characteristic time 

scale hints at cooperative effects, and we have to ask if and under which circumstances 

correlations might qualitatively affect the asymptotic long-time


power laws. For according to Smoluchowski theory [12], we would expect the 

annihilation reactions to produce depletion zones in sufficiently low dimensions 

d ≤ d c , which would in turn induce a considerable slowing down of the 

density decay, see Subsect. 7.2.3. For two-species pair annihilation A + B →∅ 

(without mixing), another complication emerges, namely particle species segregation 

in dimensions for d ≤ d s ; the regions dominated by either species 

become largely inert, and the annihilation reactions are confined to rather 

sharp fronts [12]. 

Competition between particle decay and production processes, e.g., in the 

reactions A →∅(with rate κ), A ⇋ A + A (with forward and back rates σ 

and λ, respectively), leads to even richer scenarios, as can already be inferred 

from the associated rate equation 

ȧ(t) =(σ − κ) a(t) − λa(t) 2 . (7.152) 

For σκ, we encounter an 

active state with a(t) → a ∞ =(σ − κ)/λ exponentially, with rate ∼ σ − κ. We 

have thus identified a nonequilibrium continuous phase transition at σ c = κ. 

Indeed, as in equilibrium critical phenomena, the critical point is governed by 

characteristic power laws; for example, the asymptotic particle density a ∞ ∼ 

(σ−σ c ) β , and the critical density decay a(t) ∼ (λt) −α with β 0 =1=α 0 in the 

mean-field approximation. The following natural questions then arise: What 

are the critical exponents once statistical fluctuations are properly included 

in the analysis? Can we, as in equilibrium systems, identify and characterise 

certain universality classes, and which microscopic or overall, global features 

determine them and their critical dimension? 

Already the previous set of reactions may also be viewed as a (crude) model 

for the population dynamics of a single species. In the same language, we may 

also formulate a stochastic version of the classic Lotka–Volterra predator–prey 

competition model [46]: if by themselves, the “predators” A die out according 

to A →∅, with rate κ, whereas the prey reproduce B → B + B with rate σ, 

and thus proliferate with a Malthusian population explosion. The predators 

are kept alive and the prey under control through predation, here modeled as 

the reaction A + B → A + A: with rate λ, a prey is “eaten” by a predator, 

who simultaneously produces an offspring. The coupled kinetic rate equations 

for this system read 

ȧ(t) =λa(t) b(t) − κa(t) , ḃ(t) =σb(t) − λa(t) b(t) . (7.153) 

It is straightforward to show that the quantity K(t) =λ[a(t)+b(t)]−σ ln a(t)− 

κ ln b(t) is a constant of motion for this coupled system of differential equations, 

i.e., ˙K(t) = 0. As a consequence, the system is governed by regular 

population oscillations, whose frequency and amplitude are fully determined


by the initial conditions. Clearly, this is not a very realistic feature (albeit 

mathematically appealing), and moreover Eqs. (7.153) are known to be quite 

unstable with respect to model modifications [46]. Indeed, if one includes 

spatial degrees of freedom and takes account of the full stochasticity of the 

processes involved, the system’s behaviour turns out to be much richer [47]: 

In the species coexistence phase, one encounters for sufficiently large values of 

the predation rate an incessant sequence of “pursuit and evasion” waves that 

form quite complex dynamical patterns. In finite systems, these induce erratic 

population oscillations whose features are however independent of the initial 

configuration, but whose amplitude vanishes in the thermodynamic limit. 

Moreover, if locally the prey “carrying capacity” is limited (corresponding to 

restricting the maximum site occupation number per site on a lattice), there 

appears an extinction threshold for the predator population that separates the 

absorbing state of a system filled with prey from the active coexistence regime 

through a continuous phase transition [47]. 

These examples all call for a systematic approach to include stochastic 

fluctuations in the mathematical description of interacting reaction–diffusion 

systems that would be conducive to the application of field-theoretic tools, and 

thus allow us to bring the powerful machinery of the dynamic renormalisation 

group to bear on these problems. In the following, we shall describe such a 

general method [48–50] which allows a representation of the classical master 

equation in terms of a coherent-state path integral and its subsequent analysis 

by means of the RG (for overviews, see Refs. [10–12]). 

7.2.2 Field Theory Representation of Master Equations 

The above interacting particle systems, when defined on a d-dimensional lattice 

with sites i, are fully characterised by the set of occupation integer numbers 

n i =0, 1, 2,... for each particle species. The master equation then describes 

the temporal evolution of the configurational probability distribution 

P ({n i }; t) through a balance of gain and loss terms. For example, for the binary 

annihilation and coagulation reactions A + A →∅with rate λ and A + A → A 

with rate λ ′ , the master equation on a specific site i reads 

∂P(n i ; t) 

∂t 

= λ (n i +2)(n i +1)P (...,n i +2,...; t) 

+λ ′ (n i +1)n i P (...,n i +1,...; t) 

−(λ + λ ′ ) n i (n i − 1) P (...,n i ,...; t) , (7.154) 

with initially P ({n i }, 0) = ∏ i P (n i), e.g., a Poisson distribution P (n i ) = 

¯n ni 

0 e−¯n0 /n i !. Since the reactions all change the site occupation numbers by 

integer values, a second-quantised Fock space representation is particularly 

useful [48–50]. To this end, we introduce the bosonic operator algebra 

] [ ] [ ] 

[a i ,a j =0= a † i ,a† j , a i ,a † j = δ ij . (7.155)


From these commutation relations one establishes in the standard manner 

that a i and a † i constitute lowering and raising ladder operators, from which 

we may construct the particle number eigenstates |n i 〉, 

a i |n i 〉 = n i |n i − 1〉 , a † i |n i〉 = |n i +1〉 , a † i a i |n i 〉 = n i |n i 〉 . (7.156) 

(Notice that we have chosen a different normalisation than in ordinary quantum 

mechanics.) A state with n i particles on sites i is then obtained from the 

empty vaccum state |0〉, defined through a i |0〉 = 0, as the product state 

|{n i }〉 = ∏ ( ) 

a † ni 

i |0〉 . (7.157) 

i 

To make contact with the time-dependent configuration probability, we 

introduce the formal state vector 

|Φ(t)〉 = ∑ {n i} 

P ({n i }; t) |{n i }〉 , (7.158) 

whereupon the linear time evolution according to the master equation is translated 

into an “imaginary-time” Schrödinger equation 

∂|Φ(t)〉 

∂t 

= −H |Φ(t)〉 , |Φ(t)〉 =e −Ht |Φ(0)〉 . (7.159) 

The stochastic quasi-Hamiltonian (rather, the time evolution or Liouville operator) 

for the on-site reaction processes is a sum of local terms, H reac = 

∑ 

i H i(a † i ,a i); e.g., for the binary annihilation and coagulation reactions, 

( 

H i (a † i ,a i)=−λ 1 − a † 2 ) 

i a 2 i − λ ′( ) 

1 − a † i a † i a2 i . (7.160) 

The two contributions for each process may be physically interpreted as follows: 

The first term corresponds to the actual process under consideration, 

and describes how many particles are annihilated and (re-)created in each 

reaction. The second term gives the “order” of each reaction, i.e., the number 

operator a † i a i appears to the kth power, but in normal-ordered form as 

a † k 

i a 

k 

i ,forakth-order process. Note that the reaction Hamiltonians such as 

(7.160) are non-Hermitean, reflecting the particle creations and destructions. 

In a similar manner, hopping between neighbouring sites 〈ij〉 is represented 

in this formalism through 

( 

) 

a † i − a† j 

)(a i − a j . (7.161) 

H diff = D ∑ 〈ij〉 

Our goal is of course to compute averages with respect to the configurational 

probability distribution P ({n i }; t); this is achieved by means of the 

projection state 〈P| = 〈0| ∏ i eai , which satisfies 〈P|0〉 =1and〈P|a † i = 〈P|,


since [e ai ,a † j ]=eai δ ij . For the desired statistical averages of observables that 

must be expressible in terms of the occupation numbers {n i }, we then obtain 

〈F (t)〉 = ∑ {n i} 

F ({n i }) P ({n i }; t) =〈P| F ({a † i a i}) |Φ(t)〉 . (7.162) 

Let first us explore the consequences of probability conservation, i.e., 1 = 

〈P|Φ(t)〉 = 〈P|e −Ht |Φ(0)〉. This requires 〈P|H = 0; upon commuting e ∑ i ai 

with H, effectively the creation operators become shifted a † i → 1+a† i , whence 

this condition is fulfilled provided H i (a † i → 1,a i) = 0, which is indeed satisfied 

by our explicit expressions (7.160) and (7.161). By this prescription, we may 

also in averages replace a † i a i → a i , i.e., the particle density becomes a(t) = 

〈a i 〉, and the two-point operator a † i a i a † j a j → a i δ ij + a i a j . 

In the bosonic operator representation above, we have assumed that there 

exist no restrictions on the particle occupation numbers n i on each site. If, 

however, there is a maximum n i ≤ 2s+1, one may instead employ a representation 

in terms of spin s operators. For example, particle exclusion systems 

with n i = 0 or 1 can thus be mapped onto non-Hermitean spin 1/2 “quantum” 

systems. Specifically in one dimension, such representations in terms of integrable 

spin chains have proved a fruitful tool; for overviews, see Refs. [51–54]. 

An alternative approach uses the bosonic theory, but encodes the site occupation 

restrictions through appropriate exponentials in the number operators 

e −a† i ai [55]. 

We may now follow an established route in quantum many-particle theory 

[56] and proceed towards a field theory representation through constructing 

the path integral equivalent to the “Schrödinger” dynamics (7.159) based 

on coherent states, which are right eigenstates of the annihilation operator, 

a i |φ i 〉 = φ i |φ i 〉, with complex eigenvalues φ i . Explicitly, one finds 

( 

|φ i 〉 =exp − 1 ) 

2 |φ i| 2 + φ i a † i |0〉 , (7.163) 

satisfying the overlap and (over-)completeness relations 

〈φ j |φ i 〉 =exp 

(− 1 2 |φ i| 2 − 1 ) ∫ ∏ 

2 |φ j| 2 + φ ∗ d 2 φ i 

j φ i , 

π |{φ i}〉〈{φ i }| =1. 

i 

(7.164) 

Upon splitting the temporal evolution (7.159) into infinitesimal steps, and 

inserting Eq. (7.164) at each time step, standard procedures (elaborated in 

detail in Ref. [12]) yield eventually 

∫ ∏ 

〈F (t)〉 ∝ 

i 

D[φ i ] D[φ ∗ i ] F ({φ i })e −A[φ∗ i ,φi] , (7.165) 

with the action

A[φ ∗ i ,φ i ]= ∑ i 


( ∫ tf 

[ 

] ) 

−φ i (t f )+ dt φ ∗ ∂φ i 

i 

0 ∂t + H i(φ ∗ i ,φ i ) −¯n 0 φ ∗ i (0) , (7.166) 

where the first term originates from the projection state, and the last one from 

the initial Poisson distribution. Notice that in the Hamiltonian, the creation 

and annihilation operators a † i and a i are simply replaced with the complex 

numbers φ ∗ i and φ i, respectively. 

Taking the continuum limit, φ i (t) → ψ(x,t), φ ∗ i (t) → ˆψ(x,t), the “bulk” 

part of the action becomes 

∫ ∫ [ ( ) 

] 

A[ ˆψ, ∂ 

ψ] = d d x dt ˆψ 

∂t − D ∇2 ψ + H reac ( ˆψ, ψ) , (7.167) 

where the hopping term (7.161) has naturally turned into a diffusion propagator. 

We have thus arrived at a microscopic stochastic field theory for reaction– 

diffusion processes, with no assumptions whatsoever on the form of the (internal) 

noise. This is a crucial ingredient for nonequilibrium dynamics, and 

we may now use Eq. (7.167) as a basis for systematic coarse-graining and the 

renormalisation group analysis. Returning to our example of pair annihilation 

and coagulation, the reaction part of the action (7.167) reads 

H reac ( ˆψ, 

( 

ψ) =−λ 1 − ˆψ 2) ( ˆψ) 

ψ 2 − λ ′ 1 − ˆψψ 2 , (7.168) 

see Eq. (7.160). Let us have a look at the classical field equations, namely 

δA/δψ = 0, which is always solved by ˆψ = 1, reflecting probability conservation, 

and δA/δ ˆψ = 0, which, upon inserting ˆψ = 1 gives here 

∂ψ(x,t) 

= D ∇ 2 ψ(x,t) − (2λ + λ ′ ) ψ(x,t) 2 , (7.169) 

∂t 

i.e., essentially the mean-field rate equation for the local particle density 

ψ(x,t), see Eq. (7.149), supplemented with diffusion. The field theory action 

(7.167), derived from the master equation (7.154), then provides a means 

of including fluctuations in our analysis. 

Before we proceed with this program, it is instructive to perform a shift 

in the field ˆψ about the mean-field solution, ˆψ(x, t) =1+˜ψ(x, t), whereupon 

the reaction Hamiltonian density (7.168) becomes 

H reac ( ˜ψ, ψ) =(2λ + λ ′ ) ˜ψψ 2 +(λ + λ ′ ) ˜ψ 2 ψ 2 . (7.170) 

In addition to the diffusion propagator, the annihilation and coagulation 

processes thus give identical three- and four-point vertices; aside from nonuniversal 

amplitudes, one should therefore obtain identical scaling behaviour 

for both binary reactions in the asymptotic long-time limit [57]. Lastly, we 

remark that if we interpret the action A[ ˜ψ, ψ] as a response functional (7.34), 

despite the fields ˜ψ not being purely imaginary, our field theory becomes formally 

equivalent to a “Langevin” equation, wherein additive noise is added to


Eq. (7.169), albeit with negative correlator L[ψ] =−(λ + λ ′ ) ψ 2 , which represents 

“imaginary” multiplicative noise. This Langevin description is thus not 

well-defined; however, one may render the noise correlator positive through 

a nonlinear Cole–Hopf transformation ˜ψ =e˜ρ , ψ = ρ e −˜ρ such that ˜ψψ = ρ, 

with Jacobian 1, but at the expense of “diffusion noise” ∝ Dρ(∇˜ρ ) 2 in the action 

[58]. In summary, binary (and higher-order) annihilation and coagulation 

processes cannot be cast into a Langevin framework in any simple manner. 

7.2.3 Diffusion-Limited Single-Species Annihilation Processes 

We begin by analysing diffusion-limited single-species annihilation kA →∅ 

[57, 59]. The corresponding field theory action (7.167) reads 

∫ ∫ [ ( ) 

A[ ˆψ, ∂ 

ψ] = d d x dt ˆψ 

∂t − D∇2 ψ − λ 

(1 − ˆψ k) ] 

ψ k , (7.171) 

which for k ≥ 3 allows no (obvious) equivalent Langevin description. Straightforward 

power counting gives the scaling dimension for the annihilation rate, 

[λ] =µ 2−(k−1)d , which suggests the upper critical dimension d c (k) =2/(k−1). 

Thus we expect mean-field behaviour ∼ (λt) −1/(k−1) , see Eq. (7.151), in any 

physical dimension for k>3, logarithmic corrections at d c =1fork =3and 

at d c =2fork = 2, and nonclassical power laws for pair annihilation only in 

one dimension. The field theory defined by the action (7.171) has two vertices, 

the “annihilation” sink with k incoming lines only, and the “scattering” vertex 

with k incoming and k outgoing lines. Neither allows for propagator renormalisation, 

hence the model remains massless with exact scaling exponents 

η =0andz = 2, i.e., diffusive dynamics. 

In addition, the entire perturbation expansion for the renormalisation for 

the annihilation vertices is merely a geometric series of the one-loop diagram, 

see Fig. 7.6 for the pair annihilation case (k = 2). If we define the renormalised 

effective coupling according to 

g R = Z g 

λ 

D B kd µ −2(1−d/dc) , (7.172) 

where B kd = k! Γ (2−d/d c ) d c /k d/2 (4π) d/dc , we obtain for the single nontrivial 

renormalisation constant 

+ + + 

. . . 

+ 

. . . 

+ 

. . . 

Fig. 7.6. Vertex renormalisation for diffusion-limited binary annihilation A+A →∅


Z −1 

g 

=1+ λB kd µ −2(1−d/dc) 

D (d c − d) 

(7.173) 

to all orders. Consequently, the associated RG beta function becomes 

β g = µ ∂ 

∂µ ∣ g R = − 2 g R 

(d − d c + g R ) , (7.174) 

0 

d c 

with the Gaussian fixed point g ∗ 0 = 0 stable for d > d c (k) = 2/(k − 1), 

leading to the mean-field power laws (7.151), whereas for d


difference of particle numbers (even locally), i.e., there is a conservation law 

for c(t) =a(t) − b(t) =c(0) [61]. The rate equations for the concentrations 

ȧ(t) =−λa(t) b(t) =ḃ(t) (7.181) 

are for equal initial densities a(0) = b(0) solved by the single-species pair 

annihilation mean-field power law 

a(t) =b(t) ∼ (λt) −1 , (7.182) 

whereas for unequal initial densities c(0) = a(0) − b(0) > 0, say, the majority 

species a(t) → a ∞ = c(0) > 0ast →∞, and the minority population disappears, 

b(t) → 0. From Eq. (7.181) we obtain for d>d c = 2 the exponential 

approach 

a(t) − a ∞ ∼ b(t) ∼ e −c(0) λt . (7.183) 

Mapping the associated master equation onto a continuum field theory 

(7.167), the reaction term now reads (with the fields ψ and ϕ representing the 

A and B particles, respectively) [62] 

H reac ( ˆψ, 

( 

ψ, ˆϕ, ϕ) =−λ 1 − ˆψ 

) 

ˆϕ ψϕ . (7.184) 

As in the single-species case, there is no propagator renormalisation, and 

moreover the Feynman diagrams for the renormalised reaction vertex are of 

precisely the same form as for A + A →∅, see Fig. 7.6. Thus, for unequal 

initial densities, c(0) > 0, the mean-field power law ∼ λt in the exponent of 

Eq. (7.183) becomes again replaced with (Dt) d/2 in dimensions d ≤ d c =2, 

leading to stretched exponential time dependence, 

d


∫ 

c(x,t)= d d x ′ G 0 (x − x ′ ,t) c(x ′ , 0) . (7.188) 

Let us furthermore assume a Poisson distribution for the initial density correlations 

(indicated by an overbar), a(x, 0) a(x ′ , 0) = a(0) 2 + a(0) δ(x − x ′ )= 

b(x, 0) b(x ′ , 0) and a(x, 0) b(x ′ (0) = a(0) 2 , which implies c(x, 0) c(x ′ , 0) = 

2 a(0)δ(x−x ′ ). Averaging over the initial conditions then yields with Eq. (7.188) 

∫ 

c(x,t) 2 =2a(0) d d x ′ G 0 (x − x ′ ,t) 2 =2a(0) (8πDt) −d/2 ; (7.189) 

since the distribution for c will be a Gaussian, we thus obtain for the local 

density excess originating in a random initial fluctuation, 

√ √ 

2 a(0) 

|c(x,t)| = 

π c(x,t)2 =2 

π (8πDt)−d/4 . (7.190) 

In dimensions d2and∼t −d/2 for d


and D + A →∅. We may then obviously identify A = C and B = D, which 

leads back to the case of two-species pair annihilation. Generally, within essentially 

mean-field theory one finds for cyclic multi-species annihilation processes 

a i (t) ∼ t −α(q,d) , where for 

{ 

4 q =2, 4, 6,... 

2


[λ] =µ 2−d ,isthusirrelevant in the RG sense, and can be dropped for the 

computation of universal, asymptotic scaling properties. 

The action (7.195) with λ =0isknownasReggeon field theory [65], and 

its basic characteristic is its invariance under rapidity inversion S(x,t) ↔ 

− ˜S(x, −t). If we interpret Eq. (7.195) as a response functional, we see that it 

becomes formally equivalent to a stochastic process with multiplicative noise 

(〈ζ(x,t)〉 = 0) captured by the Langevin equation [66, 67] 

∂S(x,t) 

= −D ( r − ∇ 2) S(x,t) − uS(x,t) 2 + ζ(x,t) , 

∂t 

(7.196) 

〈ζ(x,t) ζ(x ′ ,t ′ )〉 =2uS(x,t) δ(x − x ′ ) δ(t − t ′ ) (7.197) 

(for a more accurate mapping procedure, see Ref. [68]), which is essentially a 

noisy Fisher–Kolmogorov equation (7.193), with the noise correlator (7.197) 

ensuring that the fluctuations indeed cease in the absorbing state where 

〈S〉 = 0. It has moreover been established [69–71] that the action (7.195) describes 

the scaling properties of critical directed percolation (DP) clusters [72], 

illustrated in Fig. 7.7. Indeed, if the DP growth direction is labeled as “time” 

t, we see that the structure of the DP clusters emerges from the basic decay, 

branching, and coagulation reactions encoded in Eq. (7.194). 

t ⃗ 

Fig. 7.7. Directed percolation process (left) and critical DP cluster (right) 

In fact, the field theory action should govern the scaling properties of 

generic continuous nonequilibrium phase transitions from active to inactive, 

absorbing states, namely for an order parameter with Markovian stochastic 

dynamics that is decoupled from any other slow variable, and in the absence 

of quenched randomness [71, 73]. This DP conjecture follows from the following 

phenomenological approach [68] to simple epidemic processes (SEP), or 

epidemics with recovery [46]: 

1. A “susceptible” medium becomes locally “infected”, depending on the density 

n of neighboring “sick” individuals. The infected regions recover after 

a brief time interval. 

2. The state n =0isabsorbing. It describes the extinction of the “disease’. 

3. The disease spreads out diffusively via the short-range infection 1. of neighboring 

susceptible regions.


4. Microscopic fast degrees of freedom are incorporated as local noise or 

stochastic forces that respect statement 2., i.e., the noise alone cannot 

regenerate the disease. 

These ingredients are captured by the coarse-grained mesoscopic Langevin 

equation ∂ t n = D ( ∇ 2 − R[n] ) n + ζ with a reaction functional R[n], and 

s stochastic noise correlator of the form L[n] =nN[n]. Near the extinction 

threshold, we may expand R[n] =r + un + ..., N[n] =v + ..., and higherorder 

terms turn out to be irrelevant in the RG sense. Upon rescaling, we 

recover the Reggeon field theory action (7.195) for DP as the corresponding 

response functional (7.34). 

We now proceed to an explicit evaluation of the DP critical exponents 

to one-loop order, closely following the recipes given in Subsects. 7.1.3–7.1.5. 

The lowest-order fluctuation contribution to the two-point vertex function 

Γ (1,1) (q,ω) (propagator self-energy) is depicted in Fig. 7.8(a). The Feynman 

rules of Subsect. 7.1.3 yield the corresponding analytic expression 

Γ (1,1) (q,ω)=iω + D (r + q 2 )+ u2 

D 

∫ 

k 

1 

iω/2D + r + q 2 /4+k 2 . (7.198) 

The criticality condition Γ (1,1) (0, 0) = 0 at r = r c 

fluctuation-induced shift of the percolation threshold 

provides us with the 

∫ Λ 

r c = − u2 1 

D 2 r c + k 2 + O(u4 ) . (7.199) 

k 

Inserting τ = r − r c into Eq. (7.198), we then find to this order 

∫ 

Γ (1,1) (q,ω)=iω + D (τ + q 2 ) − u2 iω/2D + τ + q 2 /4 

D k 2 (iω/2D + τ + q 2 /4+k 2 ) , (7.200) 

k 

and the diagram in Fig. 7.8(b) for the three-point vertex functions, evaluated 

at zero external wavevectors and frequencies, gives 

( 

) 

Γ (1,2) ({0}) =−Γ (2,1) ({0}) =−2u 1 − 2u2 1 

D 

∫k 

2 (τ + k 2 ) 2 . (7.201) 

For the renormalisation factors, we use again the conventions (7.56) and 

(7.58), but with 

(a) 

(b) 

Fig. 7.8. DP renormalisation: one-loop diagrams for the vertex functions (a) Γ (1,1) 

(propagator self-energy), and (b) Γ (1,2) = −Γ (2,1) (nonlinear vertices)


u R = Z u uA 1/2 

d 

µ (d−4)/2 . (7.202) 

Because of rapidity inversion invariance, Z ˜S 

= Z S . With Eq. (7.53) the derivatives 

of Γ (1,1) with respect to ω, q 2 ,andτ, as well as the one-loop result for 

Γ (1,2) in Eq. (7.201), all evaluated at the normalisation point τ R =1,provide 

us with the Z factors 

Z S =1− u2 A d µ −ɛ 

2D 2 ɛ 

Z τ =1− 3u2 A d µ −ɛ 

4D 2 ɛ 

From these we infer the RG flow functions 

, Z D =1+ u2 A d µ −ɛ 

4D 2 ɛ 

, Z u =1− 5u2 A d µ −ɛ 

4D 2 ɛ 

, 

. (7.203) 

γ S = v R /2 , γ D = −v R /4 , γ τ = −2+3v R /4 , (7.204) 

with the renormalised effective coupling 

v R = Z2 u 

Z 2 D 

whose RG beta function is to this order 

u 2 

D 2 A d µ d−4 , (7.205) 

β v = v R 

[ 

−ɛ +3vR + O(v 2 R) ] . (7.206) 

For d>d c = 4, the Gaussian fixed point v ∗ 0 = 0 is stable, and we recover 

the mean-field critical exponents. For ɛ =4− d>0, we find the nontrivial 

IR-stable RG fixed point 

v ∗ = ɛ/3+O(ɛ 2 ) . (7.207) 

Setting up and solving the RG equation (7.65) for the vertex function 

proceeds just as in Subsect. 7.1.5. With the identifications (7.83) (with a =0) 

we thus obtain the DP critical exponents to first order in ɛ, 

η = − ɛ 6 + O(ɛ2 ) , 

1 

ν =2− ɛ 4 + O(ɛ2 ) , 

z =2− ɛ 

12 + O(ɛ2 ) . (7.208) 

In the vicinity of v ∗ , the solution of the RG equation for the order parameter 

reads, recalling that [S] =µ d/2 , 

) 

〈S R (τ R ,t)〉 ≈µ d/2 l (d−γ∗ S )/2 Ŝ 

(τ R l γ∗ τ ,v 

∗ 

R ,D R µ 2 l 2+γ∗ D t , (7.209) 

which leads to the following scaling relations and explicit exponent values, 

β = 

ν(d + η) 

2 

=1− ɛ 6 + O(ɛ2 ) , α = β 

zν =1− ɛ 4 + O(ɛ2 ) . (7.210) 

The scaling exponents for critical directed percolation are known analytically 

for a plethora of physical quantities (but the reader should beware that


various different conventions are used in the literature); for the two-loop results 

to order ɛ 2 in the perturbative dimensional expansion, see Ref. [68]. In 

Table 7.2, we compare the O(ɛ) values with the results from Monte Carlo 

computer simulations, which allow the DP critical exponents to be measured 

to high precision (for recent overviews on simulation results for DP and other 

absorbing state phase transitions, see Refs. [74, 75]). Yet unfortunately, there 

are to date hardly any real experiments that would confirm the DP conjecture 

[71, 73] and actually measure the scaling exponents for this prominent 

nonequilibrium universality class. 

Table 7.2. Comparison of the DP critical exponent values from Monte Carlo simulations 

with the results from the ɛ expansion 

Scaling exponent d =1 d =2 d =4− ɛ 

ξ ∼|τ| −ν ν ≈ 1.100 ν ≈ 0.735 ν =1/2+ɛ/16 + O(ɛ 2 ) 

t c ∼ ξ z ∼|τ| −zν z ≈ 1.576 z ≈ 1.73 z =2− ɛ/12 + O(ɛ 2 ) 

a ∞ ∼|τ| β β ≈ 0.2765 β ≈ 0.584 β =1− ɛ/6+O(ɛ 2 ) 

a c(t) ∼ t −α α ≈ 0.160 α ≈ 0.46 α =1− ɛ/4+O(ɛ 2 ) 

7.2.6 Dynamic Isotropic Percolation and Multi-Species Variants 

An interesting variant of active to absorbing state phase transitions emerges 

when we modify the SEP rules (1) and (2) in Subsect. 7.2.5 to 

1. The susceptible medium becomes infected, depending on the densities n 

and m of sick individuals and the “debris”, respectively. After a brief time 

interval, the sick individuals decay into immune debris, which ultimately 

stops the disease locally by exhausting the supply of susceptible regions. 

2. The states with n = 0 and any spatial distribution of m are absorbing, 

and describe the extinction of the disease. 

Here, the debris is given by the accumulated decay products, 

∫ t 

m(x,t)=κ n(x,t ′ )dt ′ . (7.211) 

−∞ 

After rescaling, this general epidemic process (GEP) or epidemic with removal 

[46] is described in terms of the mesoscopic Langevin equation [76] 

∂S(x,t) 

= −D ( r − ∇ 2) ∫ t 

S(x,t) − DuS(x,t) S(x,t ′ ) Dt ′ + ζ(x,t) , 

∂t 

−∞ 

(7.212) 

with noise correlator (7.197). The associated response functional reads [77,78]


∫ ∫ [ ( 

A[ ˜S,S]= 

∂ 

d d x dt ˜S 

∂t + D ( r − ∇ 2)) t 

] 

S − u ˜S 2 S + DuS∫ 

S(t ′ ) . 

(7.213) 

For the field theory thus defined, one may take the quasistatic limit by 

introducing the fields 

˜ϕ(x) = ˜S(x,t→∞) , ϕ(x) =D 

∫ ∞ 

−∞ 

S(x,t ′ )dt ′ . (7.214) 

For t →∞, the action (7.213) thus becomes 

∫ [ 

] 

A qst [˜ϕ, ϕ] = d d x ˜ϕ r − ∇ 2 − u (˜ϕ − ϕ) ϕ, (7.215) 

which is known to describe the critical exponents of isotropic percolation [79]. 

An isotropic percolation cluster is shown in Fig. 7.9, to be contrasted with 

the anisotropic scaling evident in Fig. 7.7(b). The upper critical dimension of 

isotropic percolation is d c = 6, and an explicit calculation, with the diagrams 

of Fig. 7.8, but involving the static propagators G 0 (q) =1/(r +q 2 ), yields the 

following critical exponents for isotropic percolation, to first order in ɛ =6−d, 

η = − ɛ 

21 + O(ɛ2 ) , 

1 

ν =2− 5 ɛ 

21 + O(ɛ2 ) , β =1− ɛ 7 + O(ɛ2 ) . (7.216) 

In order to calculate the dynamic critical exponent for this dynamic isotropic 

percolation (dIP) universality class, we must return to the full action (7.213). 

Once again, with the diagrams of Fig. 7.8, but now involving a temporally 

nonlocal three-point vertex, one then arrives at 

z =2− ɛ 6 + O(ɛ2 ) . (7.217) 

For a variety of two-loop results, the reader is referred to Ref. [68]. It is also 

possible to describe the crossover from isotropic to directed percolation within 

this field-theoretic framework [80, 81]. 

Fig. 7.9. Isotropic percolation cluster


Let us next consider multi-species variants of directed percolation processes, 

which can be obtained in the particle language by coupling the DP reactions 

A i →∅, A i ⇋ A i + A i via processes of the form A i ⇋ A j + A j (with j ≠ i); 

or directly by the corresponding generalisation within the Langevin representation 

with 〈ζ i (x,t)〉 =0, 

∂S i 

∂t = D ( 

i ∇ 2 − R i [S i ] ) S i + ζ i , R i [S i ]=r i + ∑ j 

g ij S j + ... , (7.218) 

〈ζ i (x,t)ζ j (x ′ ,t ′ )〉=2S i N i [S i ] δ(x − x ′ ) δ(t − t ′ ) δ ij ,N i [S i ]=u i + ... (7.219) 

The ensuing renormalisation factors turn out to be precisely as for singlespecies 

DP, and consequently the generical critical behaviour even in such 

multi-species systems is governed by the DP universality class [58]. For example, 

the predator extinction threshold for the stochastic Lotka–Volterra 

system mentioned in Subsect. 7.2.1 is characterised by the DP exponents as 

well [47]. But these reactions also generate A i → A j , causing additional terms 

∑ 

j≠i g j S j in Eq. (7.218). Asymptotically, the inter-species couplings become 

unidirectional, which allows for the appearance of special multicritical points 

when several r i = 0 simultaneously [82]. This leads to a hierarchy of order 

parameter exponents β k on the kth level of a unidirectional cascade, with 

β 1 =1− ɛ 6 + O(ɛ2 ) , 

β 2 = 1 2 − 13 ɛ 

96 + O(ɛ2 ) ,..., β k = 1 − O(ɛ) ; (7.220) 

2k for the associated crossover exponent, one can show Φ =1toall orders [58]. 

Quite analogous features emerge for multi-species dIP processes [58, 68]. 

7.2.7 Concluding Remarks 

In these lecture notes, I have described how stochastic processes can be 

mapped onto field theory representations, starting either from a mesoscopic 

Langevin equation for the coarse-grained densities of the relevant order parameter 

fields and conserved quantities, or from a more microsopic master 

equation for interacting particle systems. The dynamic renormalisation group 

method can then be employed to study and characterise the universal scaling 

behaviour near continuous phase transitions both in and far from thermal 

equilibrium, and for systems that generically display scale-invariant features. 

While the critical dynamics near equilibrium phase transitions has been thoroughly 

investigated experimentally in the past three decades, regrettably such 

direct experimental verification of the by now considerable amount of theoretical 

work on nonequilibrium systems is largely amiss. In this respect, applications 

of the expertise gained in the nonequilibrium statistical mechanics 

of complex cooperative behaviour to biological systems might prove fruitful 

and constitutes a promising venture. One must bear in mind, however, that 

nonuniversal features are often crucial for the relevant questions in biology.


In part, the lack of clearcut experimental evidence may be due to the fact 

that asymptotic universal properties are perhaps less prominent in accessible 

nonequilibrium systems, owing to long crossover times. Yet fluctuations do 

tend to play a more important role in systems that are driven away from thermal 

equilibrium, and the concept of universality classes, despite the undoubtedly 

much increased richness in dynamical systems, should still be useful. For 

example, we have seen that the directed percolation universality class quite 

generically describes the critical properties of phase transitions from active 

to inactive, absorbing states, which abound in nature. The few exceptions 

to this rule either require the coupling to another conserved mode [83, 84]; 

the presence, on a mesoscopic level, of additional symmetries that preclude 

the spontaneous decay A → ∅ as in the so-called parity-conserving (PC) 

universality class, represented by branching and annihilating random walks 

A → (n +1)A with n even, andA + A →∅[85] (for recent developments 

based on nonperturbative RG approaches, see Ref. [86]); or the absence of any 

first-order reactions, as in the (by now rather notorious) pair contact process 

with diffusion (PCPD) [87], which has so far eluded a successful field-theoretic 

treatment [88]. A possible explanation for the fact that DP exponents have not 

been measured ubiquitously (yet) could be the instability towards quenched 

disorder in the reaction rates [89]. 

In reaction–diffusion systems, a complete classification of the scaling properties 

in multi-species systems remains incomplete, aside from pair annihilation 

and DP-like processes, and still constitutes a quite formidable program 

(for a recent overview over the present situation from a field-theoretic viewpoint, 

see Ref. [12]). This is even more evident for nonequilibrium systems in 

general, even when maintained in driven steady states. Field-theoretic methods 

and the dynamic renormalisation group represent powerful tools that I 

believe will continue to crucially complement exact solutions (usually of onedimensional 

models), other approximative approaches, and computer simulations, 

in our quest to further elucidate the intriguing cooperative behaviour 

of strongly interacting and fluctuating many-particle systems. 

Acknowledgements 

This work has been supported in part by the U.S. National Science Foundation 

through grant NSF DMR-0308548. I am indebted to many colleagues, 

students, and friends, from and with whom I had the pleasure to learn and 

research the material presented here; specifically I would like to mention 

Vamsi Akkineni, Michael Bulenda, John Cardy, Olivier Deloubrière, Daniel 

Fisher, Reinhard Folk, Erwin Frey, Ivan Georgiev, Yadin Goldschmidt, Peter 

Grassberger, Henk Hilhorst, Haye Hinrichsen, Martin Howard, Terry Hwa, 

Hannes Janssen, Bernhard Kaufmann, Mauro Mobilia, David Nelson, Beth 

Reid, Zoltán Rácz, Jaime Santos, Beate Schmittmann, Franz Schwabl, Steffen 

Trimper, Ben Vollmayr-Lee, Mark Washenberger, Fred van Wijland, and 

Royce Zia. Lastly, I would like to thank the organisers of the very enjoyable


Luxembourg Summer School on Ageing and the Glass Transition for their 

kind invitation, and my colleagues at the Laboratoire de Physique Théorique, 

Université de Paris-Sud Orsay, France and at the Rudolf Peierls Centre for 

Theoretical Physics, University of Oxford, U.K., where these lecture notes 

were conceived and written, for their warm hospitality. 

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